Discrete time and Aristotle's argument for an infinite past

Aristotle had a famous argument that time had no beginning or end. In the case of beginnings, this argument caused immense philosophical suffering in the middle ages, since combined with the idea that time requires change it implies that the universe was eternal, contrary to the Jewish, Muslim and Christian that God created the universe a finite amount of time ago.

The argument is a reductio ad absurdum and can be put for instance like this:

1. Suppose t₀ is the beginning of time.

2. Before t₀ there is no time.

3. It is a contradiction to talk of what happened before the the beginning of time.

4. But if (1) is true, then (2) talks of what is before the beginning of time.

5. Contradiction!

It’s pretty easy to see what’s wrong with the argument. Claim (2) should be charitably read as:

• Not (before t₀ there is time).

Seen that way, (2) doesn’t talk about what happened before t₀, but is just a denial that there was any such thing as time-before-t₀.

It just struck me that a similar argument could be used to establish something that Aristotle himself rejects. Aristotle famously believed that time was discrete. But now argue:

6. Suppose t₀ and t₁ are two successive instants of time.

7. After t₀ and before t₁ there is no time.

8. It is a contradiction of what happened when there is no time.

9. But if (7) is true, then (7) talks of what is when there is no time.

10. Contradiction!

Again, the problem is the same. We should take (7) to deny that there is any such thing as time-after-t₀-and-before-t₁.

So Aristotle needed to choose between his preference for the discreteness of time and his argument for an infinite past.

What if there is no tomorrow?

There are two parts of Aristotle’s theory that are hard to fit together.

First, we have Aristotle’s view of future contingents, on which

1. It is neither true nor false that tomorrow there will be a sea battle

but, of course:

2. It is true that tomorrow there will be a sea battle or no sea battle.

Of course, nothing rides on “tomorrow” in (1) and (2): any future metric interval of times will do. Thus:

3. It is true that in 86,400,000 milliseconds there will be a sea battle or not.

(Here I adopt the convention that “in x units” denotes the interval of time corresponding to the displayed number of significant digits in x. Thus, “in 86,400,000 ms” means “at a time between 86,399,999.5 (inclusive) and 86,400,000.5 (exclusive) ms from now.”)

Second, we have Aristotle’s view of time, on which time is infinitely divisible but not infinitely divided. Times correspond to what one might call happenings, the beginnings and ends of processes of change. Now which happenings there will be, and when they will fall with respect to metric time (say, 3.74 seconds after some other happening), is presumably something that is, or can be, contingent.

In particular, in a world full of contingency and with slow-moving processes of change, it is contingent whether there will be a time in 86,400,000 ms. But (3) entails that there will be such a time, since if there is no such time, then it is not true that anything will be the case in 86,400,000 ms, since there will be no such time.

Thus, Aristotle cannot uphold (3) in a world full of contingency and slow processes. Hence, (3) cannot be a matter of temporal logic, and thus neither can (2) be, since logic doesn’t care about the difference between days and milliseconds.

If we want to make the point in our world, we would need units smaller than milliseconds. Maybe Planck times will work.

Objection: Suppose that no moment of time will occur in exactly x₁ seconds, because x₁ falls between all the endpoints of processes of change. But perhaps we can still say what is happening in x₁ seconds. Thus, if there are x₀ < x₁ < x₂ such that x₀ seconds from now and x₂ seconds from now (imagine all this paragraph being said in one moment!) are both real moments of time, we can say things about what will happen in x₁ seconds. If I will be sitting in both x₀ and x₂ seconds, maybe I can say that I will be sitting in x₁ seconds. Similarly, if Themistocles is leading a sea battle in 86,399,999 ms and is leading a sea battle in 86,400,001 ms, then we can say that he is leading a sea battle in 86,400,000 ms, even though there is no moment of time then. And if he won’t lead a sea battle in either 86,399,999 ms or in 86,400,000 ms, neither will he lead one in 86,400,000 ms.

Response: Yes, but (3) is supposed to be true as a matter of logic. And it’s logically possible that Themistocles leads a sea battle in 86,399,999 ms but not in 86,400,001 ms, in which case if there will be no moment in 86,400,000 ms, we cannot meaningfully say if he will be leading a sea battle then or not. So we cannot save (3) as a matter of logic.

A possible solution: Perhaps Aristotle should just replace (2) with:

4. It is true that will be: no tomorrow or tomorrow a sea battle or tomorrow no sea battle.

I am a bit worried about the "will" attached to a “no tomorrow”. Maybe more on that later.

Presentism, multiverses and discrete time

Suppose time is in fact continuous and modeled by the real numbers.

It seems odd indeed to me that the real numbers should be the only possible way for time to run. The real numbers are a very specific mathematical system. There are other systems, such as the hyperreals or the rationals or even the integers, that seem to be plausible alternatives. I know of no argument that the time sequence has to be numbered by the real numbers.

Thus, given our initial supposition, it should be possible to have time sequences corresponding to ordered sequences numbered by the integers or the hyperreals. Here, then, is a further intuition. It is possible to have a multiverse with radically different spacetime structures in each universe of it. If so, then we would expect the possibility of a multiverse where different universes in a multiverse have time sequences based on very different ordered sets.

Suppose presentism is necessarily true. Then even in such a multiverse, there would be an absolute present running across all of these timelines in the different universes. And that would be rather odd. Imagine that in one universe the time-line is corresponds to the integers and in the other it corresponds to the reals, and both are found in one multiverse. What happens in the universe whose time-line is based on the integers when the line of the present moves continuously across the uncountable infinity of times numbered by the real numbers? Does it stay for infinitely moments at the same integer? But then at infinitely many moments of time it would be at one moment, which is a contradiction. Or does the universe with the integer time-line pop out of existence when the present doesn’t meet up with these integers? Maybe that’s the best view, but it’s a weird view.

Perhaps the presentist’s best bet is to say that there is a privileged mathematical structure that models what a time-line could be like. If so, my intuition says that the only candidate for that privileged structure would be a discrete structure like the integers. For there are arguments in the history of philosophy for time having to be discrete (arguments from Zeno through myself), but none for time having to be modeled by specifically the real numbers, or the rational numbers, or some specific hyperreal field.

Information Processing Finitism

When I was trying to work out my intuitions about causal paradoxes of infinity, which eventually led to my formulating the thesis of causal finitism (CF)—that nothing can have an infinite causal history—I toyed with views that involved information. I ended up largely abandoning that approach, partly because of my qualms about the concept of information and perhaps partly because of worries about physics that I will discuss below.

But I still think the alternative, which one might call information processing finitism, is something someone should work out in more detail.

• [IPF] Nothing with finite informational content can essentially causally depend on anything with infinite informational content.

Here, informational content is by definition contingent. The “essentially” excludes cases where finite informational content depends on a finite part of something with infinite informational content. How exactly the “essentially” is spelled out is one thing I am not clear on as yet.

The main difficulty with IPF is that our physics seems to violate it. The exact current temperature in Waco depends on the exact temperature, pressure and other facts around the world yesterday. Each of the latter facts involves infinite information—temperature is quantified with a real number, and a real number contains infinite information. Note that here IPF and CF may diverge. An advocate of CF can say that the exact current temperature in Waco depends on a finite number of past events such as “yesterday particle n has parameters P”, even if the parameters P involve real numbers that have infinite energy.

One way to escape this difficulty is to assume that our fundamental physics is actually discrete, and the real numbers in our equations are just an approximation. But I don’t want to stick my neck out so far.

Let’s see if we can make IPF work out with a continuous dynamics. We can suppose that metaphysically speaking, an entity’s having a real-valued parameter is constituted by the entity’s having an infinite sequence of discrete parameters, which parameters are more ontologically fundamental than the real-valued parameter.

For instance, by a one-to-one mapping we can assume our real number is strictly between zero and one, and then define it as an infinite decimal sequence 0.b₁b₂..., specified by an infinite sequence of digits. Unfortunately, then, we have some severe restrictions on what kind of dynamics we can have if we require that each digit of the output depend only on a finite number of digits of the input. For instance, multiplication by 3/4 cannot be defined, because to know whether f(x) starts with 0.24 or 0.25, you’d have to know whether x < 1/3 or x ≥ 1/3, and if the input is 0.333..., then you can’t tell from a finite number of digits which is the case. This kind of problem will occur with any other base.

It would be really nice to find some way of encoding a real number as an infinite sequence of discrete parameters each of which takes on a fixed finite range that escapes this kind of a problem. I am pretty sure this is impossible, but am too tired to prove it right now.

But there is another approach. We can have non-unique (many-to-one) encodings of reals. Here is one such approach, probably not the most natural one. Consider sequences of natural numbers n₁, n₂, ... such that for all k we have n_k ≤ 2k and there exists a real number x between 0 and 1 inclusive with the property that |x−n_k/2k| ≤ 1/k. Say that such a sequence encodes the real number x. In general, there will be more than one sequence encoding x by this rule.

Then if f is a function from [0,1] to [0,1], if we have a sequence n₁, n₂, ... encoding the real number x, to generate an acceptable kth term in a sequence encoding f(x), it suffices to know f(x) to within precision 1/2k, and if f is continuous, then we can do that by knowing a finite number of terms in a sequence encoding x (this is because every continuos function on [0,1] is uniformly continuous).

So any continuous dynamics from [0,1] to [0,1] can be handled in this way. The cost is that fundamental reality has degrees of freedom that are unimportant physically—for fundamental reality distinguishes between different sequences encoding the same real x, but the difference has no physical significance.

I don’t know if there is a way to do this with a unique encoding.

The fleetingness of being

Imagine it’s the last moment of time. What’s next for you? Nothing! It’s a terrifying time, but it’s one that’s hard to describe well. Phrases like “You’re about to perish” don’t fit it logically, because they imply that you will perish, but at the last moment of time there is no “will”. You need awkward wide-scope negations like: “It is not the case that you will continue to exist.”

But I think the philosophical puzzles go beyond the choice of words.

Thing about a world where time begins and ends with t₁, where there is only one moment. That’s a world with no flux or flow or dynamism or change. It seems, then, that that’s a world where essentially temporal attitudes, like fear of ceasing to exist, are inappropriate. It doesn’t, it seems, to be a world where it’s right for you to feel the terror of facing nothingness. Indeed, it doesn’t seem like anything in this world is fleeting or lasting.

But the difference between the only-one-moment and last-moment scenarios is just with regard to the past. Now in the last-moment scenario you would have a reasonable (pace Epicurus) terror of impending nonexistence and a vivid feeling of the fleetingness of existence.

But taking away the past, and hence moving to the only-on-moment scenario, shouldn’t change any of that! It doesn’t make your existence last any longer. It makes you no more eternal. We have to be able to say that somehow in the only-one-moment world our existence would be tenuous and fleeting (indeed, it seems, maximally so).

This pulls us to a very deep conclusion here:

1. The phenomenon of fleetingness does not require the flow of time.

For in the only-one-moment world we have fleetingness but no flow.

So if we are to look at what grounds the fleetingness of our existence, it seems we must look away from the distinctive resources of the A-theory of time, and towards the B-theory.

One obvious thing to say is that there is an incompleteness to our existence when restricted to any finite compass. Eighty years is not enough for the kind of being we are, and a moment is much less. This is something an eternalist can say, whether or not they accept the A-theory or the B-theory of time. Though it’s not quite so clear that a presentist or Growing Blocker can say it, since on their views our future life is not a part of reality anyway, no matter whether it is finite or infinite.

But perhaps there is a resource available for the A-theorist, even the presentist. Instead of thinking that it is the present moment that is present, we can suppose that what is present is an interval between two succeeding times in a discrete account of time. If so, then neither the only-one-moment and last-moment scenarios work. Instead, one has only-one-interval and last-interval scenarios. And these are not so problematic. Even if there is only one interval of time, that’s enough for change and flow—things move from one state to another over an interval. The impending doom has to do with the fact that the later end of the present interval borders nothingness. And over that interval, we can say (if we have a flowy theory of time) that we are flowing—but not for long!

Of course, there are technical issues with the suggestion that what is present is an interval between two successive times. If there is flow during that interval, it sees can always ask: “How long before the interval is finished?” But any clear answer to that subdivides the interval and places us at a moment within it. So we must refuse to countenance any answer beyond: “I am flowing from t_n to t_n + 1.” (We might then say: We’re between 0 and t_n + 1 − t_n units of time before the next interval begins.)

I started thinking about an A-theory on which what is present is an interval just as an exercise in wacky theories of time. I am now thinking that perhaps this is the best version of presentism.

Two implications of Aristotle's theory of time and locality

Aristotle thinks that time is infinitely subdivisible but only finitely subdivided. Thus, there will be moments t₁ < t₂ such that there is actually no moment between t₁ and t₂, but there could have been. What would make there have been a moment between t₁ and t₂? Presumably, this would be if something happened strictly between those times. Time is the measure of change, so if, say, some object started or finished changing at a time between t₁ and t₂, then there would have been a time between them, say t_1.5.

But here is a curious consequence. Suppose that in the actual world, w₀, I am living from t₁ to t₂, which are so close together in time that there are no time between them. But in another world, t₁, where everything in our galaxy was the same, in some other galaxy indeterministically an event happened between t₁ and t₂, namely at t_1.5. Then:

• In w₀, it is not true that I exist at t_1.5 (because there is no t_1.5).

• In w₁, it is true that I exist at t_1.5.

And what is responsible for that difference is that indeterministic event in another galaxy. So it seems that something in another galaxy is responsible, in a faster-than-light way, for whether I exist at t_1.5. In other words, the Aristotelian theory seems to imply highly non-local influences.

There is perhaps a way out. Perhaps fundamentally time sequences are internal to substances. Thus, I have a time sequence internal to me, you have one internal to you, and things in that other galaxy have time sequences internal to them. There are, additionally, connections (probably causal ones) between objects that allow one to form a global time sequence. That global time sequence will include moments that don’t correspond to any moments internal to me. For instance, it will include moments earlier than my conception, but more interestingly, it could be that for me t₂ immediately succeeds t₁, but something else has a time that fits between t₁ and t₂, and so global time could have times corresponding to t₁ and t₂, but also some intermediate time between them.

The difference between w₀ and w₁, then, would not be a difference in my internal time sequence. What happened in that other galaxy wouldn’t affect my internal time except perhaps once the light from that galaxy could reach me.

On this account, while it is true that I exist at t_1.5, my existing at t_1.5 is not an intrinsic feature of me. The difference between my existing at t_1.5 in w₁ and my not existing at t_1.5 in w₀ is a merely Cambridge difference.

I think it is hard to make this story fit with presentism. When t_1.5 is present, then it had better be intrinsic to me that I exist presently, i.e., at t_1.5. A similar point applies to growing block.

Maybe, though, there is a way of making this story fit with a moving spotlight A-theory. We could suppose that at global time t_1.5, what is “lit up” by the spotlight is the time t_1.5 for the thing in the other galaxy that has something happening to it then, but for me what is lit up is the entire interval between t₁ and t₂.

If I am right, then

1. Locality, and

2. Aristotle’s theory of time

seem to imply:

3. Internal time is primary

4. Eternalism is true.

Another weird discrete theory of time

Suppose time is discrete. The usual story then is that we are always at some point of time. But what if, instead, we are always between times? I.e., the times themselves are something like imaginary points—we don’t occupy them on their own. It is only the interval between two successive times that we occupy, and we occupy the interval as a whole. Such an interval is a “now”.

If t_n and t_n + 1 are successive times, then we say that at (t_n,t_n + 1) (think of this as an ordered pair or an interval—your choice of mathematical representation!):

• x is F iff x is F at t_n and at t_n + 1

• x is non-F iff x is non-F at t_n and at t_n + 1

• x exists iff x exists at t_n and at t_n + 1

• x non-exists iff x is does not exist at t_n or at t_n + 1

• x is changing from F to non-F iff x is F at t_n but not at t_n + 1

• x is changing from non-F to F iff x is non-F at t_n and at t_n + 1

• x is coming into existence iff x exists at t_n + 1 but not at t_n

• x is ceasing to exist iff x exists at t_n but not at t_n + 1

• x is coming to be F iff x is F at t_n + 1 and either does not exist at t_n or exists at t_n but is not F then

• x is ceasing to be F iff x is F at t_n and either does not exist at t_n + 1 or exists at t_n + 1 but is not F then.

Here is a plausible thesis:

1. x fails to exist or x is F or x is non-F.

On the theory we are exploring, this is false in a now. Instead:

2. x non-exists or is coming into existence or is ceasing to exist or is F or is non-F or is changing from F to non-F or is changing from non-F to F.

This theory is a variant of one I tried out in an earlier post, minus the possibility of the now being a point.

Simultaneous causation and determinism

Consider the Causal Simultaneity Thesis (CST) that all causation is simultaneous. Assume that simultaneity is absolute (rather than relative). Assume there is change. Here is a consequence I will argue for: determinism is false. In fact, more strongly, there are no diachronic deterministic causal series. What is surprising is that we get this consequence without any considerations of free will or quantum mechanics.

Since there is a very plausible argument from presentism to CST (a non-simultaneous fundamental causal relation could never obtain between two existent things given presentism), we get an argument from presentism to indeterminism.

Personally, I am inclined to think of this argument as a bit of evidence against CST and hence against presentism, because it seems to me that there could be a deterministic world, even though there isn’t. But tastes differ.

Now the argument for the central thesis. The idea is simple. On CST, as soon as the deterministic causes of an effect are in place, their effect is in place. Any delay in the effect would mean a violation of the determinism. There can be nothing in the deterministic causes to explain how much delay happens, because all the causes work simultaneously. And so if determinism is true—i.e., if everything has a deterministic cause—then all the effects happen all at once, and everything is already in the final state at the first moment of time. Thus there is no change if we have determinism and CST.

The point becomes clearer when we think about how it is an adherent of CST explains diachronic causal series. We have an item A that starts existing at time t₁, persists through time t₂ (kept in existence not by its own causal power, as that would require a diachronic causal relation, but either by a conserver or a principle of existential inertia), then causes an item B, which then persists through time t₃ and then causes an item C, and so on. While any two successive items in the causal series A, B, C, ... must overlap temporally (i.e., there must be a time at which they both exist), we need not have temporal overlap between A and C, say. We can thus have things perishing and new things coming into being after them.

But if the causation is deterministic, then as soon as A exists, it will cause B, which will cause C, and so on, thereby forcing the whole series to exist at once, and destroying change.

In an earlier post, I thought this made for a serious objection to CST. I asked: “Why does A ‘wait’ until t₂ to cause B?” But once we realize that the issue above has to do with determinism, we see that an answer is available. All we need to do is to suppose there is probabilistic causation.

For simplicity (and because this is what fits best with causal finitism) suppose time is discrete. Then we may suppose that at each moment of time at which A exists it has a certain low probability p_AB of causing B if B does not already exist. Then the probability that A will cause B precisely after n units of time is (1 − p_AB)ⁿp_AB. It follows mathematically that “on average” it will cause B after p_AB/(1 − p_AB) fundamental units of time.

It follows that for any desired average time delay, a designer of the universe can design a cause that has that delay. Let’s say that we want B to come into existence on average u fundamental units of time after A has come into existence. Then the designer can give A a causal power of producing B at any given moment of time at which B does not already exist with probability p_AB = 1/(1 + u).

The resulting setup will be indeterministic, and in particular we can expect significant random variation in how long it takes to get B from A. But if the designer wants more precise timing, that can be arranged as well. Let’s say that our designer wants B to happen very close to precisely one second after A. The designer can then ensure that, say, there are a million instants of time in a second, and that A has the power to produce an event A₁ with a probability at any given instant such that the expected wait time will be 0.0001 seconds (i.e., 100 fundamental units of time), and A₁ the power to produce A₂ with the same probability, and so on, with A₁₀₀₀₀ = B. Then by the Central Limit Theorem, the average wait time between A and B can be expected to be fairly close to 10000 × 0.0001 = 1 seconds, and the designer can get arbitrarily high confidence of an arbitrarily high precision of delay by inserting more instants in each second, and more intermediate causes between A and B, with each intermediate cause having an average delay time of 100 fundamental units (say). (This uses the fact that the geometric distribution has a finite third moment and the Barry-Esseen version of the Central Limit Theorem.)

Thus, a designer of the universe can make an arbitrarily precise and reliable near-deterministic changing universe despite CST. And that really blunts the force of my anti-deterministic observation as a consideration against CST.

Continuous choices

Suppose at at noon, Alice is relaxed in an armchair listening to music, but at any given time she is capable of choosing to get up, walk over to the kitchen and make herself a sandwich for lunch, which it’s time for. For fifteen minutes she continues listening to the music and then gets up at 12:15. It seems that she is continually responsible for her continuing to sit until 12:15, and then she is responsible for getting up.

Here is one realistic question about what happened between 12:00 and 12:15:

1. Did Alice make a vast number of choices, one at every moment until 12:15, to remain seated, and then at 12:15 a choice to get up?

In favor of a positive answer, it is difficult to see how she could be responsible for not getting up at a given time if she did not choose not to get up.

But a positive answer seems psychologically implausible. Indeed, it doesn’t seem like Alice would be enjoying the music if every moment she had to positively choose to stay.

Also, let’s think about what the reasons weighing in on each choice would be. On the one hand, there is a very weak reason to get up now. It’s a weak reason because getting up the next moment would be just as good hunger-wise. On the other hand, there is a very weak reason to keep sitting in order to enjoy music between this moment and the next. It’s a weak reason because the amount of music involved is very small. Choices on the basis of such very weak reasons are hard to make. These reasons would be hard to weigh. And when making choices between hard to weigh reasons, it seems that the chances of going for either option should be of the same order of magnitude. But if Alice were to make a vast number of choices between getting up and staying between, say, 12:00 and 12:10, with each choice having roughly the same order of magnitude of probability, then it was very unlikely that all these choices were choices to stay.

I find the responsibility argument pretty persuasive, though. Maybe, though, the right story that balances psychological plausibility with intuitions about responsibility is this: Alice made a small number of choices between 12:00 and 12:15. Most of these choices were a choice whether to think harder about whether to get up or just let the status quo continue “for a while”. Most of the time, she chose just to let the status quo roll on. At a time t during which the status quo was “just rolling on”, Alice’s responsibility for not getting up was derivative from her choice to stop thinking about the question. Sometimes, however, Alice decided to think harder about whether to get up. Finally, she thought harder, and got up.

Since the number of choices is smaller on this story, it doesn’t interfere as much with the enjoyment. There is some interference, but that’s realistic. And since the number of choices is smaller, the probabilities of each option can be of the same order of magnitude without this creating any problems.

Now, prescinding from the realism behind the discussion of (1), we can ask the also interesting question:

2. Could it be that both (a) time is continuous and (b) Alice literally makes a choice to remain seated at every single moment of time between 12:00 and 12:15?

The answer, I think, is negative. For consider a choice at t. Alice would be choosing between the good of slightly more music and the good of slightly earlier relief of hunger. But how long as the “slightly more” and “slightly earlier”? Zero temporal length! For if time is continuous, and Alice is choosing at every moment, zero length of time elapses between choices. Indeed, there is no sense to the idea of “between choices”. So Alice would be choosing between zero-value goods. And that doesn’t make rational sense.

A spatial "in between"

In my last post I offered the suggestion that someone who thinks time is discrete has reason to think that there is something in between the moments—a continuous unbroken (but perhaps breakable) interval.

I think a similar thought can be had about discrete space.

Consideration 1: Imagine that space is discrete, arranged on a grid pattern, and I touch left and right index fingers together. It could happen that the rightmost spatial points of my left fingertip is side-by-side with the leftmost spatial points of my right fingertip, but nonetheless my hands aren’t joined into a single solid. One way to represent this setup would be to say that a spatial point in my left fingertip is right next to a spatial point in my right fingertip, but the interval between these spatial points is not within me.

But positing a spatial “in between” isn’t the only solution: distinguishing internal and external geometry is another.

Consideration 2: Zeno’s Stadium argument can be read as noting that if space and time are discrete, then an object moving at one point per unit of time rightward and an equal length object moving at one point per unit of time leftward can pass by each other without ever being side-by-side. Positing an “in between”, such that objects may be “inbetween places when they are in between times, may make this less problematic.

A non-reductive eternalist theory of change

It is sometimes said that B-theorists see change as reducible to temporal variation of properties—being non-F at t₁ but F at t₂ (the “at-at theory of change”)—while A-theorists have a deeper view of change.

But isn’t the A-theorist’s view of change just something like: having been non-F but now being F? But that’s just as reductive as the B-theorist’s at-at theory of change, and it seems just as much to be a matter of temporal variation. Both approaches have this feature: they analyze change in terms of the having and not having of a property. Note, also, that the A-theorist who gives the having-been-but-now-being story about change is committed to the at-at theory being logically sufficient for change from being non-F to being F.

I think there may be something to the intuition that the at-at theory doesn’t wholly capture change. But moving to the A-theory does not by itself solve the problem. In fact, I think the B-theory can do better than the best version of the A-theory.

Let me sketch an Aristotelian story about time. Time is discrete. It has moments. But it is not exhausted by moments. In addition to moments there are intervals between moments. These intervals are in fact undivided, though they might be divisible (Aristotle will think they are). At moments, things are. Between moments, things become. Change is when at one moment t₁ something is non-F, at the next moment t₂ it is F, and during the interval between t₁ and t₂ it is changing from non-F to F.

On this story, the at-at theory gives a necessary condition for changing from non-F to F, but perhaps not a sufficient one. For suppose temporally gappy existence is possible, so that an object can cease to exist and come back. Then it is conceivable that an object exist at t₁ and at t₂, but not during the interval between t₁ and t₂. Such an object might be brought back into existence at t₂ with the property of Fness which it lacked at t₁, but it wouldn’t have changed from being non-F to being F.

But there is a serious logical difficulty with the above story: the law of excluded middle. Suppose that a banana turns from non-blue (say, yellow) to blue over the interval I from t₁ to t₂. What happens during the interval? By excluded middle, the banana is non-blue or blue. But which is it? It cannot be non-blue on a part of the interval I and blue on another part, for that would imply a subdivision of the interval on the Aristotelian view of time. So it must be blue over the whole interval or non-blue over the whole interval. But neither option seems satisfactory. The interval is when it is changing from non-blue to blue; it shouldn’t already be at either endpoint during the interval. Thus, it seems, during I the banana is neither non-blue nor blue, which seems a contradiction.

But the B-theorist has a way of blocking the contradiction. She can take one of the standard B-theoretic solutions to the problem of temporary intrinsics and use that. For instance, she can say that the banana is neither blue-during-I and nor non-blue-during-I. There is no contradiction here, nor any denial of excluded middle.

What the theory denies is temporalized excluded middle:

1. For any period of time u, either s during u or (not s) during u

but it affirms:

2. For any period of time u, either s during u or not (s during u).

A typical presentist is unable to say that. For a typical presentist thinks that if u is present, then s during u if and only if s simpliciter, so that (1) follows from (2), at least if u is present (and then, generalizing, even if it’s not). Such a typical presentism, which identifies present truth with truth simpliciter is I think the best version of the A-theory.

Thinking of time as made up of moments and intervals is, I think, quite fruitful.

Murder by slowdown?

Zeno wants Alice dead and he has the following plan. He slows down Alice’s functioning—say, by cooling her or by sending her around the earth on a spaceship so fast that relativistic time dilation does the job—so much that each second of Alice’s internal time takes a billion years of external time. In six seconds of Alice’s internal time, she’s dead, because the sun runs out of hydrogen and turns into a red giant.

Did Zeno kill Alice or did the sun kill Alice? Both: Zeno kills Alice by shifting her future life into a spatiotemporal position where that life would be destroyed by the sun. This is akin to sending Alice now into the sun on a speeding rocket.

(I am not a lawyer, but I expect Zeno could only be convicted of attempted murder, since a conviction for murder requires the victim to be dead; similarly, I assume that an 80-year-old person who gives someone a poison that takes forty years to work can only be convicted of attempted murder, because by the time the poison does its work, the murderer will be dead.)

But now imagine that Zeno lives in a universe where the earth will be habitable forever. He sets up an automated system that slows down Alice’s internal time to such a degree that in the first year of external time, Alice’s internal time moves ahead only 3 seconds; in the next external year, it moves ahead by 1.5 seconds; in the next year, it moves ahead by 0.75 seconds; and so on. What happens? Well, Alice still cannot have more than six seconds of life ahead of him. In n years of external time, she will have had 6 − 6/2ⁿ seconds of internal time.

So just as in the first scenario, Zeno has ensured that Alice has less than six seconds of internal time left. It sure sounds like murder. But wait! In the second scenario, it seems that Alice never dies: she is alive this year, just sluggish; she will be alive next year, though even more sluggish; and so on.

But Alice will be dead in exactly six seconds of internal time. So what will be the cause of death? The unfortunate misalignment between Alice’s internal time and the external time of the universe, together with the universe running out of time “once year ω rolls around”? Maybe. I am not sure. This is paradoxical.

There is a way of getting out of this paradox. Suppose internal time must be discrete. Then to slow down Alice’s time means to space out the discrete ticks of her time. Suppose for simplicity that Alice has a hundred ticks per internal second. Then in the next year, she will have 300 ticks. Some time in year ten, the 599th tick of Alice’s future life happens. And the 600th tick will never happen. So, the gradual slowdown story is is impossible. The speed hits zero after the tenth year. The best (or worst?) Zeno can do is ensure that the 599th tick of Alice’s life is the last one. But if that’s what he does, then he causes her death by ensuring that the 600th tick never happens. But if that’s what he does, there is no gradual slowdown paradox.

Pleasure and pain are discrete

Alice, Bob and Chuck each come into existence at 1 o'clock.

• Alice lives for one hour. She feels continuous and unchanging morally innocent pleasure during that hour with no pain.
• Bob lives for two hours. During the first hour his experiences are exactly like Alice's; during the second hour, these experiences re-run.
• Chuck has the same internal stream of subjective experiences as Bob, but is accelerated by a factor of two relative to external time, so he lives only for one hour.

Now, let's add some axioms about hedonic value:

1. If x and y have the same internal stream of subjective experiences, though perhaps at different external rates, their lives are hedonically equally.
2. If x and y live during the same period of external time, and at each moment experience the same pleasure or pain, their lives are hedonically equal.
3. If x and y live hedonically equal lives, and y and z live hedonically equal lives, then x and z live hedonically equal lives (hedonic equality is transitive).
4. The same pleasant experience lived twice is hedonically better than when lived once.

Note that (4) needs to be carefully understood. Of course, a longer stint of a pleasant experience can get boring. But if one experiences boredom, that's not the same experience then.

Now, we have a contradiction. For by (1), Chuck and Bob's lives are hedonically equal. But by (2), Alice's and Chuck's lives are hedonically equal. Here's why. Take any time t between 1 and 2 o'clock, i.e., any time during the lives of Alice and Chuck. Because the pleasure is constant during that hour-long period, Alice's pleasure at t is the same as her pleasure at (say) 1:30. And for the same reason, Chuck's pleasure at t is the same as his pleasure at (say) 1:15. But Chuck's state at 1:15 is the same as Bob's state at 1:30, since Chuck lives the same life that Bob does, but twice as fast. And Bob's state at 1:30 is the same as Alice's state at 1:30. So, Chuck's pleasure at t is equal to Alice's pleasure at t, and hence by (2) Alice's and Chuck's lives are hedonically equal. Hence, by transitivity, Alice's and Bob's lives are hedonically equal. And this contradicts (4).

Assuming our hedonic axioms (1)-(4) are correct, the story about Alice, Bob and Chuck leads to a contradiction. So what's wrong with the story? I think it's the assumption that it's possible for pleasure to be continuous. Instead, I submit, temporally extended pleasure has to be discrete, made up of a finite number of pieces of pleasure. These pieces might be instantaneous or temporally undivided but extended. And of course the argument can be run with pain in place of pleasure as well.

A puzzle about pain and time

Suppose that at each time at which Jim experiences a pain, Sally experiences a pain that is exactly alike phenomenally, and vice versa. Suppose also that their attitudes to this kind of pain are the same. Then with respect to pain, neither is better off than the other.

But now let's add that Jim and Sally are born in the year 2000 and both die in 2040. But Jim's life proceeds twice as fast as Sally's (due to drugs or Special Relativity), so that in one minute of external time, Jim experiences two minutes of subjective time. Let's suppose that Jim and Sally have the same attitudes towards pain, and that Jim has a constant headache from his 20th subjective year until her 30th subjective year, while Sally has the same intensity of headache from her 10th subjective year until her 15th subjective year. Then Jim suffers that headache from 2010 through 2015, while Sally suffers it from 2010 until 2015. The headache is constant, so at every time at which Jim suffers a headache, Sally suffers exactly the same headache, and vice versa. And let's suppose that's all the pain either of them suffers.

By the plausible principle I started the post with, Jim and Sally are equally well off with respect to pain. But on the other hand, Jim has ten subjective years of pain while Sally only has five. Clearly, Jim's life is worse pain-wise than Sally's, even though at every time at which Jim suffers, Sally suffers equally, and vice versa.

Here are some interesting ways out.

1. Time is discrete. Then we can't really speed up a life by a factor of two--we'd have to skip every second "frame".
2. There can't be any literally constant pains. Instead, pains register at discrete moments of mental life. If these moments are sufficiently closely spaced, the pain seems continuous. On this discrete moment theory of pain, Jim will have to have twice as many moments of pain from 2010 until 2015 as Sally does, so it's false that at every time at which Jim has a pain, so does Sally.
3. Pains attach to intervals of mental life rather than particular moments. This makes the equivalence principle I started the post with make no sense, since there is no such thing as having a pain at a particular time, just as the Zenonian teleportation argument can be read as teaching us that there is no such thing as moving at a particular time.

Teleporting Zeno's arrow

Here are some plausible theses:

1. Necessarily, an object that is in the same place at time t as it has been for some non-zero period of time prior to t is not moving at t.
2. Necessarily, if an object is at one location at t₁ and at another at t₂ is moving at some time t at one of the two times or between them.
3. It is possible to have continuous time.
4. If it is possible to have continuous time, it is possible to have continuous time and instantaneous teleportation of the following sort: an object is in one place for some time up to and including t₁, then it is instantaneously teleported to a second place where it remains at all times after t₁ up to and including t₂.

These theses are logically incompatible. For, given (3) and (4), suppose we have a world with continuous time and instantaneous teleportation like in (4). Then by (2), this object moves at some time at or between the two times. But at t₁ the object is in the same place as it has been for some time, so by (1) it's not moving. And it's also not moving at any time after t₁ (up to t₂), since at any time after t₁, it's been sitting in the second location for some time.

In some ways, this is an improved version of Zeno's arrow paradox. Zeno had an implausibly strong version of (1) that implied that an object that stayed in the same place for an instant wasn't moving at that instant. That's implausible. But (1) is much weaker. The cost of this weakening is that we need to replace run-of-the-mill movement with teleportation.

Of the premises, I think (4) is the most secure, despite being the most complex. Surely God could teleport things. Here is an argument for (1). Whether an object is in motion at t should not be a future contingent at t. But if the answer to the question whether an object is in motion at t depends on what happens after t, then it would be a future contingent. So it only depends on what happens at or before t. Now if the object has been at the same place for some time prior to t, and is there at t, it should be possible (barring special cases like where God promised that the object will move) for the object to remain there for some time after t. In that case, the object would obviously not be moving at t. But since what happens after t is irrelevant to whether it's moving at t, we conclude that as long as the object has been standing in the same place for some time up to and including t, it's not moving at t.

That leaves (2) and (3). I am inclined to reject both of them myself, though of course the argument only requires one to reject one (given the reasons to believe (1) and (4)). Rejecting (2) seems to go hand-in-hand with seeing motion as something that doesn't happen at times, but only between times (the presentist may well have trouble with this).

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.

Causation and collapse

If determinism were true, then since each state could be project from the initial state, we could simply suppose that the whole four-dimensional shebang came into existence causally "all at once", so that there would be no causal relations within the four-dimensional universe. The only relevant causation could that of God's causing the universe as a whole--and an atheist might just think the four-dimensional universe to be uncaused.

I think that this acausal picture could be adapted to give an attractive picture of the role of causation in a collapse interpretation of quantum mechanics (whether the collapse is of the GRW-type or of the consciousness-caused type). On a collapse picture, we have an alternation between a deterministic evolution governed by the Schroedinger equation and an indeterministic collapse. Why not suppose, then, that there is no causation within the deterministic evolution? We could instead suppose that the state of the universe at collapse causes the whole of the four-dimensional block between that collapse and the next. As long as collapse isn't too frequent, this could allow occasions of causation to be discrete, with only a finite number of such occasions within any interval of time. And this would let us reconcile quantum physics with causal finitism even with a continuous time. (Relativity would require more work.)

Space, time and discreteness

A first plausible thesis:

1. Space is discrete if and only if time is discrete.

This is very plausible in any picture like that given by modern physics where space and time, while not on par, are very closely related. It's also intuitive. For suppose space is continuous but time is discrete. Then a small enough object will jump over some intervening space when it moves during a unit of time, which seems strange. Conversely, if space is discrete but time is continuous, then objects always move jerkily: for a bit of time they stand still, then they instantaneously jump to a new point in space, and then they stand still for a little longer. That's strange, too.

Here's a second plausible thesis:

2. A small object can rotate by any small angle without its internal measurements changing much.

But now imagine that space is discrete, with the smallest distance between points being about α. Imagine an object occupying exactly two points x and y spaced out by approximately α. Now swivel that object slightly around the point x without changing internal measurements much. The other point in the object will either continue occupying y, in which case the object hasn't rotated (contrary to (2)), or will occupy some point very close to y (but there isn't any such point, since the minimum spacing is about α), or the object will come to occupy only one point (in which case its internal measurements will change much).

If the above argument isn't clear, just imagine a hexagonal or square grid, an object composed of two points on the grid, and think what it would be to rotate that object by a small angle (smaller than 90 degrees in the case of the square grid and smaller than 60 degrees for the hexagonal one).

So (2) gives us good reason to deny that space is discrete, and then (1) gives us good reason to deny that time is discrete.

But this was too quick. Both my arguments for (1) and my argument that (2) forbids space to be discrete made a crucial assumption, namely that space has a certain fixity to it that it is independent of the objects in space. For suppose that time is continuous but space is discrete. I said that it follows that objects move jerkily. Not so. For the points that the objects occupy could be moving with the objects! Thus an object could move smoothly because the spatial points in it could be moving. The discrete space, then, wouldn't be a regular grid. It would be a mess of points, which shift around as the objects they are in shift. (This doesn't affect the argument that we shouldn't say that space is continuous but time is discrete.)

The same flaw affects my argument based on (2). I was assuming that as I rotate the two-point object, the points x and y stay fixed. But what if points are defined by objects, and so the point y rotates with the object? Again, we wouldn't have a regular grid. We would have an irregular changing grid, where the real points are defined by the objects.

The resulting view of space would be, I think, a version of Aristotle's picture, where space is infinitely divisible but not actually infinitely divided. In the case of our two point object, there could be a point at distance α/10 from y, but there isn't, unless we rotate the object that defines the points.

In other words, Aristotle's account of space is the only discretist view of space that accommodates the intuition that objects can be rotated by small amounts without great distortion. That's pretty neat, I think.

What's the motivation for thinking this is the truth of the matter. Well, causal finitism gives one good reason to think that time is discrete (or at least discrete when we restrict ourselves to a local area of space). The implication from discrete time to discrete space in (1) survives my above criticism of the argument. So we have good reason to think space is discrete. And then the rotation argument yields a version of Aristotle's view.