On two knowledge arguments

There is a structural similarity between the main reasons for adopting the A-theory of time and the knowledge argument against physicalism. In both cases, it is claimed that there is some information that we know but which is left out of the reductive theory:

1. I know that Alice is sitting now

and

2. I know what it feels like to be sitting.

The first piece of knowledge cannot be derived from data about tenseless reality and the second cannot be derived from data about physical reality, or so it is claimed.

The similarity between the two arguments suggests that there should be a correlation between dualism and adherence to the A-theory of time: for if one is convinced by one argument, one is more likely to be convinced by the other, and if one is unconvinced by one, one is less likely to be convinced by the other. Speaking for myself, I am a B-theorist dualist, and while I am unconvinced by the time argument, I go back and forth on the mind one.

It is interesting, though, to see if we can go beyond superficial similarity. One way to do that is to see if the best responses to one of the arguments can generate plausible responses to the other.

The best response to the time argument seems to be the Kaplan story that “now” is a mere indexical, and that the content of “Alice is sitting now” is the proposition that Alice sitting is at t₁ (if t₁ is now), though the character or linguistic meaning of “Alice is sitting now” is something different from the character of “Alice is sitting at t₁” (specifically, a character is a function from world-utterance pairs to propositions, and this character assigns to an utterance of “Alice is sitting now” at t in w the proposition that Alice is sitting at t).

Is there a similar story about mind argument? It’s not so clear to me. Perhaps a start would be to say that what makes me it true that I know what it feels like to be sitting is that:

3. I know that sitting feels like this.

The physicalist analogue to the Kaplan story would then be that “Sitting feels like this” expresses the proposition that sitting feels like ϕ where ϕ is some physical state of affairs, but the character or linguistic meaning of “Sitting feels like this” and “Sitting feels like ϕ” are different. I don’t think this works, however. There are two ways of taking this approach:

1. ϕ is a specific neural state that I have when I feel like I’m sitting (say, S-fibers firing)

or

2. ϕ is a complex functional state that anything has when it feels like it’s sitting, a state implemented by different neural or other physical states in different beings.

On (a), we have an analogy to the time case, for we can take the character of “Sitting feels like this” to be a function that assigns to world-utterance pair the proposition that sitting feels like ϕ where ϕ is the physical state that is the feeling for the utterer in that world. But there is also a serious disanalogy: for in the time case, the B-theorist knows (or can claim to know) the character, since the B-theorist knows a priori the specific rule by which a referent is assigned to “now” at a world-utterance pair. But the physicalist does not know a priori the specific functional story which assigns a referent to “like this” at a world-utterance pair.

On (b), we have a disanalogy, since the character is constant: at every world-utterance pair, the same proposition is assigned as the content of “Sitting feels like this.”

Still, maybe there is still a fundamental analogy, in that the time case teaches us (if we accept the Kaplan story) that one proposition can be expressed by two sentences s₁ and s₂ such that it is correct to say “I know s₁” but not correct to say “I know s₂”. Thus, I know that I am sitting now but I don’t know that I am sitting at t₁. And similarly, maybe, I know that sitting feels like this but I don’t know that sitting feels like ϕ.

What if we go the other way around, and see if the best answer to the mind argument helps with the time argument?

I guess what is generally thought to be the best answer to the mind argument is something like this: there is a conceptual difference between the “like this” of the feeling and the physical or functional state ϕ, but ontologically they are the same. And this seems very close to Michelle Beer’s defense of the B-theory.

Another prominent answer to the mind argument is to deny that the knowledge claim expresses factual knowledge, as opposed to something like know-how or imaginative mirroring. It seems to me that a know-how story could be told about the time argument: to know that Alice is sitting now is to have certain kinds of know-how concerning dealing with Alice’s sitting. The “imaginative mirroring” case might be harder.

Theories of time and truth-supervenes-on-being

Truth supervenes on being is the thesis that if two worlds have the same entities, they are otherwise the same. I just realized something that should be pretty obvious. One cannot hold on to all three of the following:

• A-theory

• eternalism

• truth supervenes on being.

For according to eternalism, at any two different times, the facts about what exists are the same. So if truth supervenes on being, at any two different times, all facts are the same—and in particular the facts about what time is objectively present will be the same, which contradicts A-theory.

In other words, just as the best version of presentism (that of Trenton Merricks) rejects that truth supervenes on being, so does the best version of the moving spotlight theory. Moreover, closed-future growing blockers—and, in particular, classical theist growing blockers—will also want to reject that truth supervenes on being since substantive truths about the future won’t supervene on being given growing block.

All this suggests that we are left with only two major theories of time available to those who accept that truth supervenes on being:

• B-theoretic eternalism

• growing block with an open future.

More on bilocation and movement

It is often said that the four-dimensionalist doesn’t have a good theory of movement beyond the at-at theory which holds that

1. to move is to be at x₁ at one time and at x₂ at a different time, where x₂ ≠ x₁.

However, I am inclined to think the at-at theory is false due to an argument that my son came up with: if an object is bilocated at both x₁ and x₂ at one time and stays unmoved in both locations until a later time, then it is true that the object is at x₁ at one time and at x₂ at another time, and yet has not moved.

It is interesting that this argument also works against the most natural tensed theory of movement, namely that:

2. an object has moved provided that it was at x₁ and is at x₂, where x₂ ≠ x₁.

For imagine that an object was and still is bilocated between x₁ and x₂ and has remained entirely unmoving. Nonetheless, it was at x₁ and is now at x₂, and x₂ ≠ x₁, so according to (2) it has moved.

Thus, my son’s argument against the at-at theory does not seem to confer an advantage on the A-theory of time.

It is tempting to tweak (2) to something like this:

3. an object has moved provided that the set of locations at which it is now present is different from a set of locations at which it was present.

But that fails. For cessation of bilocation is not movement. If an object was bilocated between two locations x₁ and x₂, and then ceased to exist at x₂, while remaining at x₁, the object nonetheless did not move, even though (3) says it did.

Furthermore, space at least could be discrete. So imagine a point particle that was bilocated at two neighboring points x₁ and x₂ in space. The particle then simultaneously moved from x₁ to x₂ and from x₂ to x₁. Yet the set of points occupied by the particle was the same as it is now. So (3) says it did not move, but it did move, twice over.

I suppose one can deny the possibility of bilocation. But that is a big price to pay, I think.

I suspect that any theory of change that the A-theorist comes up with that solves this problem will also solve the problem for the four-dimensionalist.

God and the B-theory of time

1. All reality is such that it can be known perfectly from the point of view of God.

2. The point of view of God is eternal and timeless.

3. Thus, all reality is such that it can be known perfectly from an eternal and timeless point of view.

4. If all reality is such that it can be known perfectly from an eternal and timeless point of view, then the B-theory of time is true.

5. So, the B-theory of time is true.

I am not sure of premise (4), however.

My experience of temporality

This morning I find myself feeling the force of presentism. I am finding it hard to see my four-dimensional worm theory as adequately explaining why my experience only includes what I am experiencing now, instead of the whole richness of my four-dimensional life. I am also finding it difficult to satisfactorily explain the sequentiality of my experiences: that I will have different experiences from those that I have now, some of which I dread and some of which I anticipate eagerly.

When I try to write down the thoughts that make me feel the force of presentism, the force of the thoughts is largely drained. After all, to be fair, when I wrote that I have am having trouble “explaining why my experience only includes what I am experiencing now”, shouldn’t I have written: “explaining why my present experience only includes what I am experiencing now”, a triviality? And that mysterious sequentiality, is that anything beyond the fact that some of my experiences are in the future of my present experience?

The first part of the mystery is due to the chopped up nature of my consciousness on a four-dimensional view. Instead of seeing my life as a whole, as God sees it, I see it in very short (but probably not instantaneous) pieces. It is puzzling how my consciousness can be so chopped up, and yet be all mine. But we have good reason to think that this phenomenon occurs outside of temporality. Split brain patients seem to have such chopped up consciousnesses. And if consciousness is an operation of the mind’s, then on orthodox Christology, the incarnate Christ, while one person, had (and still has) two consciousnesses.

Unfortunately, both the split brains and the Incarnation are mysterious phenomena, so they don’t do much to take away the feeling of mystery about the temporal chopping up of the consciousness of my four-dimensional life. But they do make me feel that there is no good argument for presentism here.

The second part of the mystery is due to the sequentiality of the experiences. As the split brain and Incarnation cases show, the sequentiality of experiences in different spheres of consciousness is not universal. The split brain patient has two non-sequential, simultaneous spheres of consciousness. Christ has his temporal sphere (or spheres, if we take the four-dimensional view) of consciousness and his divine atemporal sphere of consciousness. But seeing the contingency of the sequentiality does not remove the mystery in the sequentiality.

It makes me feel a little better when I recall that the presentist story about the sequentiality has its own problems. If my future experiences aren’t real—on presentism they are nothing but stuff in the scope of a modal “will” operator that doesn’t satisfy the T axiom—then what am I anticipating or dreading? It seems I am just here in the present, and when I think about this, it feels just as mysterious as on four-dimensionalism what makes the future impend. Of course, the presentist can give a reductive or non-reductive account of the asymmetry between past and future, but so can the four-dimensionalist.

So what remains of this morning’s presentist feelings? Mostly this worry: Time is mysterious and our theories of time—whether eternalist or presentist—do not do justice to its mysteriousness. This is like the thought that qualia are mysterious, but when we give particular theories of them—whether materialist or dualist—it feels like something is left out.

But what if I forget about standard four-dimensionalism and presentism, and just try to see what theory of time fits with my experiences? I then find myself pulled towards a view of time I had when I was around ten years old. Reality is four-dimensional, but we travel through it. Future sufferings I dread are there, ahead of me. But I am not just a temporal part among many: there is no future self suffering future pains and enjoying future pleasures. The past and future have physical reality but it’s all zombies. As for me, I am wholly here and now. And you are wholly here and now. We travel together through the four-dimensional reality.

But these future pains and pleasures, how can they be if they are not had by me or anyone else? They are like the persisting smile of the Cheshire cat. (I wasn’t worried about this when I was ten, because I was mainly imagining myself as traveling through events, and not philosophically thinking about my changing mental states. It wasn’t a theory, but a way of thinking.) Put that way, maybe it’s not so crazy. After all, the standard Catholic view of the Eucharist is that the accidents of bread and wine exist without anything having them. So perhaps my future and past pains and pleasures exist without anyone having them—but one day I will have them.

Even this strange theory, though, does not do justice to sequentiality. What makes it be the case that I am traveling towards the future rather than towards the past?

And what about Relativity Theory? Why don’t we get out of sync with one another if we travel fast enough relative to one another? Perhaps the twin who travels at near light speed comes back to earth and meets only zombies, not real selves? That seems absurd. Maybe though the internal flow of time doesn’t work like that.

I do not think this is an attractive theory. It is the theory that best fits most of my experience of temporality, and that is a real consideration in favor of it. But it doesn’t solve the puzzle of sequentiality. I think I will stick with four-dimensionalism. For now. (!)

Asymmetric temporal attitudes and time travel

Philosophers sometimes use thought experiments concerning the asymmetry of attitudes towards future and past events as arguments for a metaphysical asymmetry between past and future. For instance, the fact that I would prefer a much larger pain in my past to a smaller pain in the future is puzzling if the past and future are metaphysically on par.

Here’s a thesis I want to offer and briefly defend:

• It is not rationally consistent to give use thought experiments in this way and to accept the possibility of backwards time travel.

The reason is quite simple: if backwards time travel is possible, our asymmetric attitudes track personal time, not objective time. If I am going to travel 100 million years back in six minutes, I will prefer a smaller pain in five minutes to a much larger pain 100 million years ago, since both of these pains will be in my personal future and only a minute of personal time apart. But the metaphysical asymmetry between past and future tracks external time, not personal time.

Times that never become present

Could there be times that are never present? At first sight, this seems a contradiction: surely, each time t is present at itself. Given the B-theory of time, this indeed is automatically true.

Not so, however, for A-theories. There is no contradiction in the growing block growing by leaps and bounds. Imagine that suddenly a whole minute is added to the growing block. The times in the middle of that minute never got to be at the leading edge of reality, and hence never got to be present, since to be present is to be at the leading edge of reality, given growing block. Or consider the moving spotlight: the spotlight could jump ahead in the spacetime manifold by a minute or an hour or a year, skipping over the intervening bits of the manifold. It's less clear whether it is possible to have times that aren't ever present given presentism. Still, Dean Zimmerman has considered an eccentric version of presentism on which there still is a four-dimensional spacetime manifold. On such a view, times could be identified with hypersurfaces in some preferred foliation, and there might be some such hypersurfaces that never become present.

So, apart from the B-theory and many versions of presentism, we have a possibility of times that are never present. Why would we want to countenance such a nutty option, though?

I can think of two reasons. The first would be to reconcile Aristotle's theory of time with many physical theories. According to Aristotle, times are endpoints of changes, and any interval of time contains at most finitely many changes, so that time is discrete. (Causal finitism might be a reason to adopt such a theory.) But in many modern physical theories, from Newton at least through Einstein, time is a continuous coordinate. One can try to reconcile the two views by supposing that time is continuous, as Newton and Einstein suppose, but that only those times which are the endpoints of changes are ever present. Aristotle then may be right that times are discrete, as long as we understand him to be speaking only about the times that matter, namely those that ever become present. The second motivation would be to have a flash ontology--an ontology on which physical things exist only during the discrete moments of quantum collapse--while softening the counterintuitive consequence that at most times the universe is empty. For we could identify the times that ever become present with the times at which a flash occurs. Then even if at most times, in the broad sense of the word "times", the universe is empty, still the universe is non-empty at all the times that matter, namely at all the times that become present.

Neither a B-theorist nor a standard presentist can suppose times that are never present. But she might still suppose something that plays a similar functional role. She could think of abstract times as numbers or as hypersurfaces in an abstract continuous manifold. Then real time could be discrete, while abstract time is continuous.

Presentism and theoretical simplicity

It's oft stated that Ockham's razor favors the B-theory over the A-theory, other things being equal. But the theoretical gain here is small: the A-theorist need only add one more thing to her ideology over what the A-theorist has, namely an absolute "now", and it wouldn't be hard to offset this loss of parsimony by explanatory gains. But I want to argue that the gain in theoretical simplicity by adopting B-theoretic eternalism over presentism is much, much larger than that. In fact, it could be one of the larger gains in theoretical simplicity in human history.

Why? Well, when we consider the simplicity of a proposed law of nature, we need to look at the law as formulated in joint-carving terms. Any law can be formulated very simply if we allow gerrymandered predicates. (Think of "grue" and "bleen".) Now, if presentism is true, then a transtemporally universally quantified statement like:

1. All electrons (ever) are negatively charged

should be seen as a conjunction of three statements:

2. All electrons have always been negatively charged, all electrons are negatively charged and all electrons will always be positively charged.

But every fundamental law of nature is transtemporally universally quantified, and even many non-fundamental laws, like the laws of chemistry and astronomy, are transtemporally universally quantified. The fundamental laws of nature, and many of the non-fundamental ones as well, look much simpler on B-theoretic eternalism. This escapes us, because we have compact formulations like (1). But if presentism is true, such compact formulations are mere shorthand for the complex formulations, and having convenient shorthand does not escape a charge of theoretical complexity.

In fact, the above story seems to give us an account of how it is that we have scientifically discovered that eternalist B-theory is true. It's not relativity theory, as some think. Rather it is that we have discovered that there are transtemporally quantified fundamental laws of nature, which are insensitive to the distinction between past, present and future and hence capable of a great theoretical simplification on the hypothesis that eternalist B-theory is true. It is the opposite of what happened with jade, where we discovered that in fact we achieve simplification by splitting jade into two natural kinds, jadeite and nephrite.

Technical notes: My paraphrase (2) fits best with something like Prior's temporal logic. A competitor to this are ersatz times, as in Crisp's theory. Ersatz time theories allow a paraphrase of (1) that seems very eternalist:

3. For all times t, at t every electron is negatively charged.

However, first, the machinery of ersatz times is complex and so while (3) looks relatively simple (it just has one extra quantifier beyond (1)), if we expand out what "times" means for the ersatzist, it becomes very complex. Moreover on standard ersatzist views, the laws of nature become disjunctive in form, and that is quite objectionable. For a standard approach is to take abstract times to be maximal consistent tensed propositions, and then to distinguish actual times as times that were, are or will be true.

Change and presentism

This post is an illustration of how widely intuitions can differ. It is widely felt by presentists that presentism is needed for there to be "real change", that the B-theory is a "static" theory. But I have the intuition that presentism endangers real change. Real change requires real difference between the past states and present states, and real difference requires the reality of the differing states. But if there are no past states, there are no real differences between past and present states, and hence no change.

Of course, a presentist can say that although a past state is unreal, there can nonetheless be a real difference between it and a real present state, just as there can be a real difference between the world of Harry Potter and our world, even though the world of Harry Potter isn't real. In a sense of "real difference" that's true, I agree. But not in the relevant sense. Change is a relation between realities.

The presentist can also insist that my line of thought is simply a case of the grounding problem for presentism, and can be resolved in a similar way. Supposing a window has just changed from being whole to being broken. Then while the past unbroken state doesn't exist, there does exist a present state of the window having been whole. I am happy to grant this present state to the presentist, but it doesn't affect the argument. For the relevant difference isn't between the window having been whole and the window being broken. For if no one broke the window, there would still have been a difference between the state of the window having been whole and the window being broken. There is always a difference between a state of something having been so and a state of its being so, but this difference isn't the difference that constitutes change.

(Incredible as it may seem to the presentist, when I try to imagine the presentist's world, I imagine an evanescent instantaneous world that therefore doesn't exist long enough for any change to take place. I am well aware that this world includes states like it was the case that the window was whole, but given presentism, these states seem to me to be modal in nature, and akin to the state of it is the case in the Harry Potter universe that magic works, and hence are not appropriate to make the world non-evanescent.)

Probably the presentist's best bet is simply to deny that real difference in my sense is needed for change. All that's needed is that something wasn't so and now is so. But if something's having been not so and its being so doesn't imply a real difference, it's not change, I feel.

Of course, the presentist feels very similarly about the B-theorist's typical at-at theory of change (change is a matter of something's being one way at one time and another way at another time): she feels that what is described isn't really change.

And this, finally, gives us the real upshot of this post. There are interesting disagreements where one side's account of a phenomenon just doesn't seem to be a description of the relevant concept to the other side--it seems to be a change of topic. These disagreements are particularly difficult to make progress in. Compare how the compatibilist's account of freedom just doesn't seem to be a description of freedom to the libertarian.

I don't have a general theory on how to make progress past such disagreement. I do have one thing I do in such cases: I try to find as many things connected with the concept in other areas of philosophy, like epistemology, philosophy of science, ethics and natural theology. And then I see which account does better more generally.

Reality of change and change of reality

B-theorists are often accused of destroying the reality of change. That's a false accusation. B-theorists may have a reductive theory of change (to change is nothing but to have a property at one time and lack it at another), but they no more deny the reality of change than people who have a reductive theory of bachelorhood (to be a bachelor is nothing but to be a never-married marriageable man) deny the reality of bachelorhood.

However, there is a charge in the vicinity that does stick. While we B-theorists believe in the reality of change, there is an important sense in which we don't believe reality changes, since what is true simpliciter is always true simpliciter. Events don't become real or cease to be real. So we can say that we believe in the reality of change but not the change of reality.

Internal time, external time, probability and disagreement

Suppose that Jim lives a normal human life from the year 2000 to the year 2100. Without looking at a clock, what probability should Jim attach to the hypothesis that an even number of minutes has elapsed from the year 2000? Surely, probability 1/2.

Sally, on the other hand, lives a somewhat odd human life from the year 2000 to the year 2066. During every even-numbered minute of her life, her mental and physical functioning is accelerated by a factor of two. She can normally notice this, because the world around her, including the second hands of clocks, seems to slow down by a factor of two. She has won many races by taking advantage of this. An even-numbered external minute subjectively takes two minutes. Suppose that Sally is now in a room where there is nothing in motion other than herself, so she can't tell whether this was a sped-up minute or not. What probability should Sally attach to the hypothesis that an even number of minutes has elapsed from the year 2000?

If we set our probabilities by objective time, then the answer is 1/2, as in Jim's case. But this seems mistaken. If we're going to assign probabilities in cases like this—and that's not clear to me—then I think we should assign 2/3. After all, subjectively speaking, 2/3 of Sally's life occurs during the even-numbered minutes.

There are a number of ways of defending the 2/3 judgment. One way would be to consider relativity theory. We could mimic the Jim-Sally situation by exploiting the twin paradox (granted, the accelerations over a period of a minute would be deadly, so we'd have to suppose that Sally has superpowers), and in that case surely the probabilities that Sally should assign should be looked at from Sally's reference frame.

Another way to defend the judgment would be to imagine a third person, Frank, who lives all the time twice as fast as normal, but during odd-numbered minutes, he is frozen unconscious for half of each second. For Frank, an even numbered minute has 60 seconds' worth of being conscious and moving, while an odd numbered minute has 30 seconds' worth of it, and external reality stutters. If Frank is in a sensory deprivation chamber where he can't tell if external reality is stuttering, then it seems better for him to assign 2/3 to its being an even-numbered minute, since he's unconscious for half of each odd-numbered one. But Frank's case doesn't seem significantly different from Sally's. (Just imagine taking the limit as the unconscious/conscious intervals get shorter and shorter.)

A third way is to think about time travel. Suppose you're on what is subjectively a long trip in a time machine, a trip that's days internal time long. And now you're asked if it's an even-numbered minute by your internal time (the time shown by your wristwatch, but not by the big clock on the time machine console, which shows external years that flick by in internal minutes). It doesn't matter how the time machine moves relative to external time. Maybe it accelerates during every even-numbered minute. Surely this doesn't matter. It's your internal time that matters.

Alright, that's enough arguing for this. So Sally should assign 2/3. But here's a funny thing. Jim and Sally then disagree on how likely it is that it's an even-numbered minute, even though it seems we can set up the case so they have the same relevant evidence as to what time it. There is something paradoxical here.

A couple of responses come to mind:

• They really have different evidence. In some way yet to be explained, their different prior life experiences are relevant evidence.
• The thesis that there cannot be rational disagreement in the face of the same evidence is true when restricted to disagreement about objective matters. But what time it is now is not an objective matter. Thus, the A-theory of time is false.
• There can be rational disagreement in the face of the same evidence.
• There are no meaningful temporally self-locating probabilities.

A Metaphysicality Index

A grad student was thinking that Platonism isn't dominant in philosophy, so I looked at the PhilPapers survey and indeed a plurality of the target faculty (39%) accepts or leans towards Platonism. Then I got to looking at how this works across various specializations: General Philosophy of Science, Philosophy of Mind, Normative Ethics, Metaethics, Philosophy of Religion, Epistemology, Metaphysics, Logic / Philosophy of Logic and Philosophy of Mathematics. And I looked at some other views: libertarianism (about free will), theism, non-physicalism about mind, and the A-theory of time.

Loosely, the five views I looked at are "metaphysical" in nature and their denials tend to be deflationary of metaphysics. I will say that someone is "metaphysical" to the extent that she answers all five questions in the positive (either outright or leaning). We can then compute a Metaphysicality Index for an individual, as the percentage of "metaphysical" answers, and then an average Metaphysicality Index per discipline.

Here's what I found. (The spreadsheet is here.) I sorted my selected M&E specialities from least to most metaphysical in the graph.

On each of the five questions, the Philosophers of Science were the least metaphysical. This is quite a remarkably un-metaphysical approach.

With the exception of Platonism, the Philosophers of Religion were the most metaphysical. (A lot of Philosophers of Religion are theists and may worry about the fit between theism and Platonism, and may think that God's ideas can do the work that Platonism is meant to do.)

Unsurprisingly, the Metaphysicians came out pretty metaphysical, though not as metaphysical as the Philosophers of Religion. (And this isn't just because the Philosophers of Religion believe in God by a large majority: even if one drops theism from the Metaphysicality Index, the Philosophers of Religion are at the top.

Interestingly, the Philosophers of Mathematics were almost as metaphysical as the Metaphysicians (average Metaphysicality Index 29.2 vs. 29.8). They were far more Platonic than anybody else. I wonder if Platonism is to Philosophy of Mathematics like Theism is to Philosophy of Religion. The Philosophers of Mathematics were also more theistic and more non-physicalistic than any group other than the Philosophers of Religion.

It's looking to me like the two fields where Platonism is most prevalent are Logic (and Philosophy of Logic) and Philosophy of Mathematics. This is interesting and significant. It suggests that on the whole people do not think one can do mathematics and logic in a nominalist setting.

For the record, here's where I stand: Platonism: no; Libertarianism: yes; God: yes; Non-physicalism: yes; A-theory: no. So my Metaphysicality Index is 60%.

The flow of time

Famously, D. C. Williams ridiculed the deep-seated intuition (at the heart of Bergson's thought, of course) that there is a flow of time, asking how fast time is flowing, if it's flowing. Some wits have tried to respond: "Always at a second per second." But there is a much better and less trivial answer. And interestingly it is an answer that has a home in the B-theory of time.

The Twin Paradox suggests that we should distinguish the internal time of an individual from something like the generally shared external time of the human community. Thinking about time travel suggests a similar distinction, as Lewis has noted. But once we have a distinction between internal and external time, then we can give a non-trivial answer to Williams' question. The flow of time is measured in terms of external units of time per internal units of time. If external time is defined by the shared life of the human community (that's one among a number of options—we should probably understand "external time" in a context-sensitive way), normally time flows at one external second per second. But if I were to engage in travel at relativistic speeds, it could be that in a month of internal time, eight years of external time would elapse. And if I were to engage in gradual backwards time travel, then I would have a negative rate of time's flow: maybe I would be moving at −1 external century per internal second. (In non-gradual time travel, the rate would be undefined.)

The distinction between internal and external time fits best with the B-theory. So the notion of time's flow, surprisingly, seems to have its home in the B-theory.

Can A-theorists believe in time travel?

I used to think that A-theorists cannot consistently believe in time travel. I think I was mistaken. As best as I can reconstruct my line of thought it was this. Time travel requires a distinction between external and internal time. If I go into a time machine, then maybe in five minutes I'll be a thousand years ago. That's a contradiction given non-circular time unless one distinguishes as follows: internally in five minutes I'll be a thousand years ago externally. But now I think that what I must have been thinking was that in five internal minutes my internal present will no longer line up with the world's objective present, since in five internal minutes my internal present will be about thousand years behind the world's external present. I don't know for sure if that's the thought I had, but if it was, it would have been a howler. For on the view, internally in five minutes, I will be at the time at which the world's external present was about a thousand years ago. Or, to put it from the external point of view, a thousand years ago I was five minutes older than I am now (age is measured internally). Even presentists can say that.

To see that this is coherent, consider a theory that takes external time to governed by the A-theory but internal time to be entirely governed by the B-theory. Thus, superimposed on the external A-series of past, present and future, there is an indexical B-series of earlier-for-me and later-for-me, where these relations are perhaps defined by internal causal relations (earlier states causing later ones). There is no more need for these two series to line up than there would be a need for the two series to line up if the external series were a B-series.

However, while this is coherent, maybe it undercuts one of the main motivations for the A-theory. For if there is a distinction between internal and external time, as there must be for time travel to be possible, all the changes we actually experience are changes with respect to internal time. In other words, they are B-type changes. But the typical A-theorist thinks B-type changes--it (internally) earlier being one way, and (internally) later another--are not what we experience when we experience "real change". Indeed, if time travel is possible, it is possible to live all of one's life at one external time, but moving through external space. Basically, just imagine that at each moment you travel to some external time t₀, but to a different spatial location in it. Maybe you have a backpack time-machine which is permanently stuck on t₀, but with the spatial locations changing. You'd experience change, because your state at internally earlier times will be different from your state at internally later times. But it would be mere B-type change, since it would all be happening at one and the same objective time.

I suppose one could say that in time-travel scenarios, especially the preceding one of living all of one's life at one external time, our experiences of change become non-veridical, for a condition on the veridicality of our experiences of change is that our internal clock lines up correctly with external time, and time-travel causes a misalignment. Maybe.

But in any case, now that we have the possibility of living all of one's life--a life that presumably could have rich causal interconnections--at one objective external time, just moving "sideways" to new spatial locations, I do think that the motivations for the A-theory decrease. For we see that what matters for the diachronic richness of our lives is that our lives be stretched over internal time, not over external time. It also matters that other people's internal times be sufficiently lined up with ours. But that doesn't call for the A-theory, either.

So, all in all, while A-theorists can believe in time travel, thinking time travel through would undercut much of the motivation for the A-theory.

McTaggart on the unreality of time

One version of McTaggart's argument for the unreality of time is:

1. If time is real, there is change.
2. If there is change, there is fundamental change, and it is the change of events or instants from future, to present, to past.
3. There is no fundamental change of events or instants from future, to present, to past.
4. So there is no change.
5. So time is not real.

(Here, "past", "present" and "future" are understood as incompatible. Thus, "past" and "future" mean wholly past and wholly future.) I am dubious of (2). But I think (3) can be defended in the following way, inspired by C. D. Broad (I think).

We need three principles:

6. If something undergoes non-Cambridge change from being F to being non-F, then at some time it exists and is F and at some time it exists and is non-F.
7. Cambridge change is never fundamental.
8. Everything that exists is present, and neither past nor future, when it exists.

Thus:

9. Nothing exhibits a non-Cambridge change from being future to being present, or from being present to being past, or from being future to being past. (6 and 8)
10. Hence nothing fundamentally changes from future, to present, to past. (7 and 9)

I think the contemporary A-theorist should deny (2). And most do.

Becoming

A-theorists talk of something called "becoming" which they say B-theorists have no room for. I don't really understand what this is. I am inclined to say that

1. x becomes F if and only if there is a time at which x is non-F and a later time at which x is F.

If so, then there is becoming—-indeed, objective becoming—-on the B-theory. Now, I think most A-theorists will agree that (1) is a necessary truth. So where's the disagreement on becoming?

Maybe, the disagreement lies in this. Although (1) is a necessary truth, nonetheless becoming is something more than just being non-F and later being F. This "something more" is necessarily there whenever something is earlier non-F and later F, but it is nonetheless something extra. (Just as God's knowing that the sky is blue is something more than just the sky being blue, but that "something more" is always there whenever the sky is blue, since God knows that the sky is blue necessarily if and only if the sky is blue.) But I really don't know what this "something more" is. I feel here like van Inwagen on substitutional quantification.

Well, that's not quite right. For it may be that (1) isn't exactly right. Suppose x has gappy existence. It exists from morning until noon and then from dinner until midnight. From morning until noon x is non-F and from dinner until midnight x is F. Does x become F? I could imagine someone saying "No" (and indeed a colleague did say just that). If not, then (1) may not be right.

Maybe one thinks gappy existence is impossible. (But why not? It seems no harder than being a spatially scattered individual, and if we're made of precisely located particles, which sure seems comprehensible, we're that.) But then consider this case. At every time that (in some unit system) is an irrational number, x is non-F and at every time that is a rational number, x is F. Then x satisfies (1). But does x ever become F? If so, when? At any time at which it is F, it also was F at an infinite number of earlier but arbitrarily close times. (This argument requires time to be a continuum, which I fear it's not.)

So one might question (1), though I am happy to bite the bullet in both cases. Maybe one should insist that to become F, you need to be non-F over an interval of times and be F over a succeeding interval of times, or something like that. But such modifications are neutral between the A- and B-theories.

Perhaps, though, the A-theorist's insistence on becoming is not so much about objects becoming a certain way, as about truths changing. The B-theorist is committed to propositions not changing in truth value. Many (but not all) A-theorists think propositions change in truth value. Thus,

2. There is becoming if and only if there is a proposition p that is true at one time and not true at another.

The B-theorist will then reject that there is becoming in this way. But notice that no it no longer seems appropriate to speak, as some A-theorists do, of doing justice to our "experience of becoming." For while we may experience the sky turning from blue to purple, we do not experience, except in a very theory-dependent way, the proposition <The sky is blue> turning from true to false. That's a change (though presumably a Cambridge one) in a Platonic entity.

"Thank goodness it's over!"

Some people think that the sentiment expressed by "Thank goodness it's over!" makes no sense apart from an A-theory of time on which there is an absolute present. But the parallel sentiment expressed by "Thank goodness this isn't happening to me!" surely had better make sense apart from some theory on which there is some "absolute I".

Change

The B-theory of time, according to which the distinctions between past, present and future (possibly unlike the distinctions between earlier-than and later-than) are merely perspectival, is often accused of being a "static theory of time".

But it is clearly a sufficient condition for x to change with respect to a predicate P that x satisfy P at one time and not at another. I am not claiming here that this is what change is. I am only claiming that satisfying a predicate at one time but not at an another is sufficient for change. How could something be round at one time and not round at another without its having changed in respect of roundness.

But of course it is a part of a typical B-theory that objects satisfy predicates at some but not at other times. In other words, something that is sufficient for change is a part of the B-theory. So how can be the B-theory be accused of being static?

Well, it could be the case that a theory T is incompatible with some phenomenon C (say, change) but nonetheless posits a phenomenon A (say, objects satisfying different predicates at different times). Such a theory is metaphysically incoherent, but of course there are metaphysically incoherent theories. So my response to the staticness charge (not the same as a static charge!) against the B-theory is not complete. But I think it shifts the onus of proof. Given that the B-theory of time posits something that clearly entails the phenomenon of change, if the theory is incompatible with the existence of change, the theory is metaphysically incoherent—and that has not been shown by its opponents. And it is too much to ask the B-theorist to prove the coherence of their theory, since showing metaphysical coherence is very hard in metaphysics. (Of course, one can prove a particular formalization of a theory to be formally coherent. And it's not hard to do that with the B-theory or the A-theory. But the question we're interested in is metaphysical coherence, not formal coherence.)

Translating A-sentences to B-sentences

I won't spell out a full method here, but I will give an example.

1. It was sunny five minutes ago.
2. It is sunny five minutes prior to t₀ (where I let t₀ be the time of this utterance).

This translation method makes use of a communicative tool that is always available to us—to stipulate new terms or symbols. Note that the parenthetical phrase is not a part of the assertion and while it is grammatically declarative, this is a declarative of stipulation rather than a declarative of assertion. In fact, the parenthetical functions exactly like prefixing with the grammatically imperative sentence: "Let t₀ be the time of this two-sentence utterance."

This avoids the standard problems with the token-reflexive method of translating A-sentences to B-sentences, such as the problem that we don't want the B-sentence to entail the existence of an utterance when the A-sentence does not. Indeed, (2) does not entail the existence of an utterance or of myself. In embeddings, the parenthetical remark functions in the wide scope. For instance:

3. It might have been that there are no speakers now.
4. It might have been that there are no speakers at t₀ (where I let t₀ be the time of this utterance).

(I don't translate "It might have been" because it is just a colloquial way of writing the box operator rather than something for a theory of time to take account of.) There is no absurd claim that there might have been no speakers at the time of this utterance, but only that there might have been no speakers at t₀.

In English, and I assume in many other natural languages, we have two ways of picking out what was said: sometimes we pick out the sentence and sometimes the proposition. Thus:

5. A: It is sunny now. B: I hope you will always remember that.
6. A: It is sunny now. B (after a pause): That's no longer true.

In (5), B is picking out A's proposition, and hoping A will remember it. (Maybe it's a romantic moment.) In (6), maybe it has started raining, and B is picking out A's sentence, and saying it's no longer true. Now, here are my translations of (5) and (6):

7. A: It is sunny at t₀ (where I let t₀ be the time of this utterance). B: At t₁, I hope that after t₁ you always remember that (where I let t₁ be the time of this utterance).
8. A: It is sunny at t₀ (where I let t₀ be the time of this utterance). B (after a pause): At t₁ a restatement of your whole sentence, including the stipulative portion, is no longer true (where I let t₁ be the time of this utterance).

The stipulative parenthesis is wide-scoped for purposes of embedding, including in "is true".

So, we can translate A-sentences into B-sentences. Since the B-theory is simpler than the A-theory, and fits better with science and theology, we should all become B-theorists at t₀ (where I let t₀ be the time of this utterance).

Why can't the past change?

If you're a B-theorist, it is no puzzle that the past can't change. It can't change because we are always in the same world, and so neither the past, nor the present nor the future can change. Today, let us suppose (I think correctly) that it is the case that on Wednesday it was raining. Could it tomorrow be the case that it wasn't raining on Wednesday? Not at all—for the very same world, the very same events, that make propositions true today is the one that we evaluate against tomorrow. The fact that the past can't change, thus, is a matter of mere logic—it just follows from the truth conditions for sentences.

But what if you're an A-theorist? So, you think that things will be objectively different tomorrow. Indeed, you already do think that some things about Wednesday will objectively change. For instance, while today (Saturday) Wednesday is objectively three days in the past, tomorrow it will objectively recede one more day into the past. So in fact we already have a change, but a change that the A-theorist doesn't mind. (Though she should.)

In any case, logic alone doesn't do the job. One way to see this is that some A-theorists actually think the future changes. Thus, today, it is false that either I am at Mass on November 8 or that I am absent from Mass on November 8. But come November 8, this disjunction will be true. But the clever tricks that open futurists use to make sense of an open future could be used, equally well, to make sense of an open past. (The parallel holds for B-theory. The B-theorist is committed to the claim that the future cannot change. This sounds fatalistic, but we must distinguish the ability to change the future from the ability to affect the future.)

In the setting of my earlier post on A-theory, the claim that the past cannot change corresponds fairly closely (and in fact exactly, if we assume a closed past) to the claim that the earlier-than relation is transitive. If today, a world where it rains on Wednesday is is past, tomorrow that world will also be past. So in the setting of that post, the explanatory challenge to the A-theorist is why the earlier-than relation E is transitive. The A-theorist who takes E to be fundamental can only say that it is a brute fact that it is necessarily transitive.

There may be A-theorists who can meet the challenge, however. Suppose that you think that there is a TimeShift operator which shifts tensed propositions time-wise. Thus, if p is the proposition that it is sunny, TimeShift(+1 day, p) is the proposition that in a day it'll be sunny. Suppose, further, we take worlds to be maximal consistent collections of propositions, or maximally specific consistent propositions. Then the TimeShift operator can also operate on worlds, and we can define E(w₁,w₂) to hold iff there is a t<0 such that w₁=TimeShift(t,w₂). Then it really is a matter of simple logic that E is transitive, and we have a perfectly good explanation of why the past cannot change.

Note, however, that an open-futurist cannot take this explanation. For her, the fixeity of the past remains a surd.

If this is right, then Tom Crisp is mistaken in taking the earlier-than relation between abstract times (which are just worlds in my terminology) to be primitive. An A-theorist should not say it's primitive—it needs explaining. Or at least I remember him taking it to be primitive, but my memory isn't so good.