Rooted and unrooted branching actualism

Branching actualist theories of modality say that metaphysical possibility is grounded in the powers of actual substances to bring about different states of affairs. There are two kinds of branching actualist theories: rooted and unrooted. On rooted theories, there are some necessarily existing items (e.g., God) whose causal powers “root” all the possibilities. On unrooted theories, we have an ungrounded infinite regress of earlier and earlier substances. In my dissertation, I defended a theistic rooted theory, but in the conclusion mentioned a weaker version on which there is no commitment to a root. At the time, I thought that not many would be attracted to an unrooted version, but when I gave talks on the material at various department, I was surprised that some atheists found the unrooted theory attractive. And such theories have indeed been more recently defended by Oppy and Malpass.

I still think a rooted version is better. I’ve been thinking about this today, and found an interesting advantage: rooted theories can allow for a tighter connection between ideal conceivability and metaphysical possibility (or, equivalently, a prioricity and metaphysical necessity). Specifically, consider the following appealing pair of connection theses:

1. If a proposition is metaphysically possible (i.e., true in a metaphysically possible world), then it is ideally conceivable.

2. If a proposition is ideally conceivable, it is true in a world structurally isomorphic to a metaphysically possible one.

The first thesis is one that, I think, fits with both the rooted and unrooted theories of metaphysical possibility. I will focus on the second thesis. This is really a family of theses, depending on what we mean by “structurally isomorphic”. I am not quite sure what I mean by it—that’s a matter for further research. But let me sketch how I’m thinking about this. A world where dogs are reptiles is ideally conceivable—it is only a posteriori that we can know that dogs are mammals; it is not something that armchair biology can reveal. A world where dogs are reptiles is metaphysically impossible. But take a conceivable but impossible world w₁ where “dogs are reptiles”—maybe it’s a world where the hair of the dogs is actually scales, and contrary to immediate appearances the dogs are cold-blooded, and so on. Now imagine a world w₂ that’s structurally isomorphic to this impossible world—for instance, all the particles are in the same place, corresponding causal relations hold, etc.—and yet where the dogs of w₁ aren’t really dogs, but a dog-like species of reptile. Properly spelled out, such a world will be possible, and denizens of that world would say “dogs are reptiles”.

Or for another example, a world w₃ where Napoleon is my child is conceivable (it’s only a posteriori that we know this world not to be actual) but impossible. But it is possible to have a world w₄ where I have a Napoleon-like child whom I name “Napoleon”. That world can be set up to be structurally isomorphic to w₃.

Roughly, the idea is this. If something is conceivable but impossible, it will become possible if we change out the identities of individuals and natural kinds, while keeping all the “structure”. I don’t know what “structure” is exactly, but I think I won’t need more than an intuitive idea for my argument. Structure doesn’t care about the identities of kinds and individuals.

Now suppose that unrooted branching actualism is true. On such a theory, there is a backwards-infinite sequence of contingent events. Let D be a complete structural description of that sequence. Let p_D be the proposition saying that some infinite initial segment of the world fits with D. According to unrooted branching actualism, p_D is actually a necessary truth. But p_D is clearly a posteriori, and hence its denial is ideally conceivable. Let w₅ be an impossible world where p_D is false. If (2) is true, then there will be a possible world w₆ which is a structural isomorph of w₅. But because p_D is a structural description, if p_D is false in a world, it is false in any structural isomorph of that world. Thus, p_D has to be false in w₆, which contradicts the assumption that p_D is a necessary truth.

The rooted branching actualist doesn’t get (2) for free. I think the only way the rooted branching actualist can accept (2) is if they think that the existence and structure of the root entities is a priori. A theist can say that: God’s existence could be a priori (as Richard Gale once suggested, maybe there is an ontological argument for the existence of God, but we’re just not smart enough to see it).

Some finitisms

I’m thinking about the kinds of finitisms there are. Here are some:

1. Ontic finitism: There can only be finitely many entities.

2. Concrete finitism: There can only be finitely many concrete entities.

3. Generic finitism: There are only finitely many possible kinds of substances.

4. Weak species finitism: No world contains infinitely many substances of a single species.

5. Strong species finitism: No species contains infinitely many possible individuals.

6. Strong human finitism: There are only finitely many possible human individuals.

7. Causal finitism: Nothing can have infinitely many items in its causal history.

8. Explanatory finitism: Nothing can have infinitely many items in its explanatory history.

I think (1) and (2) are false, because eternalism is true and it is possible to have an infinite future with a new chicken coming into existence every day.

I’ve defended (7) at length. I would love to be able to defend (8), but for reasons discussed in that book, I fear it can’t

I don’t know any reason to believe (3) other than as an implication of (1) together with realism about species. I don’t know any reason to believe (4) other than as an implication of (2) or (5).

I can imagine a combination of metaphysical views on which (6) is defensible. For instance, it might turn out that humans are made out of stuff all of whose qualities are discribable with discrete mathematics, and that there are limits on the discrete quantities (e.g., a minimum and a maximum mass of a human being) in such a way that for any finite segment of human life, there are only finitely many possibilities. If one adds to that the Principle of the Identity of Indiscernibles, in a transworld form, one will have an argument that there can only be finitely many humans. And I suppose some version of this view that applies to species more generally would give (5). That said, I doubt (6) is true.

Aquinas on per se and accidentally ordered causal series

Famously, Aquinas thinks that an accidentally ordered infinite causes is possible, but a per se ordered one is not. The difference is that in a per se ordered series ..., A₋₂, A₋₁, A₀, item A_n − 1 (for n <  − 1) is not only the cause of A_n, but is the cause of A_n’s causing of A_n + 1. But in an accidentally ordered series, A_n − 1 is not the cause of A_n’s causing of A_n + 1. Aquinas illustrates the distinction with a sequence of an infinite sequence of fathers and sons, since a grandfather is not the cause of the father’s conceiving of a son.

Now suppose we replace the people in Aquinas’s example with self-reproducing robots (von Neumann machines), each programmed by its predecessor to reproduce. Then we have a per se ordered series.

The following seems to me to be very plausible:

1. If a backwards infinite reproductive series of humans is possible, a backwards infinite reproductive series of robots is also possible.

Yet this seems to be something that Aquinas is committed to by his example of the accidentally ordered series.

Suppose one bites the bullet and denies (1). What is the relevant difference between the humans and the robots? It is presumably the determinism in the robots. Very well, then let’s suppose that each of the robots has a little hidden switch whose position is permanently set at the time of manufacturing. When the switch is in the D position, the robot is determined to reproduce at specific points in its life; when it is in the N position, at those points in its life, the robot performs an internal indeterministic quantum coin flip, reproducing on heads but not on tails.

It seems absurd to suppose that one could have a backwards infinite reproductive series of robots with the switches in the N position, but not in the D position. Yet that implausible conclusion seems to be what Aquinas’s position commits him to.

Here a suggestion for what Aquinas could do.

Aquinas thinks there is a very good metaphysical argument for rejecting backwards infinite per se ordered series. Suppose that argument is sound. Then Aquinas could say that this argument does not apply to the accidentally ordered case. But nonetheless there is a good argument based on a rearrangement principle or a principle of modal uniformity that:

2. If a backwards infinite series of robots with the switch in the N position is possible, so is a backwards infinite series of robots with the switch in the D position.

3. If a backwards infinite series of humans is possible, a backwards infinite series of robots with the switch in the N position is possible.

Given the impossibiliy of the series with the switch in the D position, it follows that the the backwards infinite sequence of humans is impossible. Aquinas can then simply say that he was wrong about his example (something that he is willing to concede anyway, due to an argument from al Ghazali specifically against an backwards infinite sequence of humans). But nothing in Aquinas’s theory commits him to the claim that every describable accidentally ordered backwards infinite sequence is possible. (An accidentally ordered backwards infinite sequence of square circles is not possible.)

At this point, Aquinas can do one of three things. First, he can say that while the backwards infinite sequence of humans or N-robots is impossible, we should remain agnostic whethere there are some backwards infinite accidentally ordered sequences are possible.

Second, he can give a plausibilistic argument that if the backwards infinite sequence of N-robots is impossible, probably all accidentally ordered backwards infinite sequences are impossible as well. (One might think this would require Aquinas to reject the possibility of an infinite past. This is not clear. He might still hold that an infinite past is possible as long as it doesn’t generate a backwards infinite causal sequence—imagine that every day in the past God creates a rock so far apart from all the other rocks that the rocks never interact).

Third, Aquinas could try to construct a new example of a backwards infinite accidentally ordered series that is possible. My intuition is that the best bet for trying to do this would be to construct a backwards infinite sequence where each item gets only a very slight causal contribution from its predecessor, and most of the explanation of the item’s existence involves God or some other single timeless being.

I myself like the second option.

A variant of Thomson's Lamp

In the classic Thomson’s Lamp paradox, the lamp has a switch such that each time you press it, it toggles between on and off. The lamp starts turned off, say, before 10:00, and then the switch is pressed at 10:00, 10:30, 10:45, 10:52.5, 10:56.25, and so on ad infinitum. And the puzzle is: Is it on or off at 11? It’s a puzzle, but not obviously a paradox.

But here’s an interesting variant. Instead of a switch that toggles on or off each time you press, you have a standard slider switch, with an off position and an on position. Before 10:00, the lamp is off. At 10:00, 10:45, 10:56.25, and so on, the switch is pushed forcefully all the way to the on side. At 10:30, 10:52.5, and so on, the switch is pushed forcefully all the way to the off side.

The difference between the slider and toggle versions is this. Intuitively, in the toggle version, each switch press is relevant to the outcome—intuitively, it reverses what the outcome would be. In the slider variant, however, each slider movement becomes irrelevant as soon as the next time happens. At 10:45, the switch is pushed to the on side, and at 10:52.5, it is pushed to the off side. But if you skipped the 10:45 push, it doesn’t matter—the 10:52.5 push ensures that the switch is off, regardless of what happened at 10:45 or earlier.

Thus, on the slider version, each of the switch slides is causally irrelevant to the outcome at 11. But now we have a plausible principle:

1. If between t₀ and t₁ a sequence of actions each of which is causally irrelevant to the state at t₁ takes place, and nothing else relevant to the state takes place, the state does not change between t₀ and t₁.

Letting t₀ be 9:59 and t₁ be 11:00, it follows from (1) that the lamp is off at 11:00 since it’s off at 10:00, since in between the lamp is subjected to a sequence of caually irrelevant actions.

Letting t₀ be 10:01 and t₁ still be 11:00, it follows from (1) that the lamp is on at 11:00, since it’s on at 10:01 and is subjected to a sequence of causally irrelevant actions.

So it’s on and off at 11:00. Now that’s a paradox!

Dry eternity

Koons and I have used causal paradoxes of infinity, such as Grim Reapers, to argue against infinite causal chains, and hence against an infinite causally-interconnected past. A couple of times people have asked me what I think of Alex Malpass’s Dry Eternity paradox, which is supposed to show that similar problems arise if you have God and an infinite future. The idea is that God is going to stop drinking (holy water, apparently!) at some point, and so he determines henceforth to act by the following rule:

1. “Every day, God will check his comprehensive knowledge of all future events to see if he will ever drink again. If he finds that he does not ever drink again, he will celebrate with his final drink. On the other hand, if he finds that his final drink is at some day in the future, he does not reward himself in any way (specifically, he does not have a drink all day).”

This leads to a contradiction. (Either there is or is not a day n such that God does not drink on any day after n. If there is such a day, then on day n + 1 God sees that he does not drink on any day after n + 1 and so by the rule God drinks on day n + 1. Contradiction! If there is no such day, then on every day n God sees that he will drink on a day later than n, and so he doesn’t drink on n, and hence he doesn’t ever drink, so that today is a day such that God does not drink on any day after it. Contradiction, again!)

Is this a problem for an infinite future? I don’t think so. For sonsider this rule.

2. On Monday, God will drink if and only if he foresees that he won’t drink on Tuesday. On Tuesday, God will drink if and only if he remembers that he drank on Monday.

Obviously, this is a rule God cannot adopt for Monday and Tuesday, since then God drinks on Monday if and only if God doesn’t drink on Monday. But this paradox doesn’t involve an infinite future, just two days.

What’s going on? Well it looks like in (2) there are two divine-knowledge-based rules—one for Monday and one for Tuesday—each of which can be adopted individually, but which cannot both be adopted, much like in (1) there are infinitely any divine-knowledge-based rules—one for each future day—any finite number of which can be adopted, but where one cannot adopt infinitely many of them.

What we learn from (2) is that there are logical limits to the ways that God can make use of divine foreknowledge. From (2), we seem to learn that one of these logical limits is that circularity needs to be avoided: a decision on Monday that depends on a decision on Tuesday and vice versa. From (1), we seem to learn that another one of these logical limits is that ungrounded decisional regresses need to be avoided: a decision that depends on a decision that depends on a decision and so on ad infinitum. This last is a divine analogue to causal finitism (the doctrine that nothing can have infinitely many things in its causal history), while what we got from (2) was a divine analogue to the rejection of causal circularity. It would be nice if there were some set of principles that would encompass both the divine and the non-divine cases. But in any case, Malpass’s clever paradox does no harm to causal finitism, and only suggests that causal finitism is a special case of a more general theory that I have yet to discover the formulation of.

Cable Guy and van Fraassen's Reflection Principle

Van Fraassen’s Reflection Principle (RP) says that if you are sure you will have a specific credence at a specific future time, you should have that credence now. To avoid easy counterexamples, the RP needs some qualifications such that there is no loss of memory, no irrationality, no suspicion of either, full knowledge of one’s own credences at any time, etc.

Suppose:

1. Time can be continuous and causal finitism is false.

2. There are non-zero infinitesimal probabilities.

Then we have an interesting argument against van Fraassen’s Reflection Principle. Start by letting RP+ be the strengthened version of RP which says that, with the same qualifications as needed for RP, if you are sure you will have at least credence r at a specific future time, then you should have at least credence r now. I claim:

3. If RP is true, so is RP+.

This is pretty intuitive. I think one can actually give a decent argument for (3) beyond its intuitiveness, and I’ll do that in the appendix to the post.

Now, let’s use Cable Guy to give a counterexample to RP+ assuming (1) and (2). Recall that in the Cable Guy (CG) paradox, you know that CG will show at one exact time uniformly randomly distributed between 8:00 and 16:00, with 8:00 excluded and 16:00 included. You want to know if CG is coming in the afternoon, which is stipulated to be between 12:00 (exclusive) and 16:00 (inclusive). You know there will come a time, say one shortly after 8:00, when CG hasn’t yet shown up. At that time, you will have evidence that CG is coming in the afternoon—the fact that they haven’t shown up between 8:00 and, say, 8:00+δ for some δ > 0 increases the probability that CG is coming in the afternoon. So even before 8:00, you know that there will come a time when your credence in the afternoon hypothesis will be higher than it is now, assuming you’re going to be rational and observing continuously (this uses (1)). But clearly before 8:00 your credence should be 1/2.

This is not yet a counterexample to RP+ for two reasons. First, there isn’t a specific time such that you know ahead of time for sure your credence will be higher than 1/2, and, second, there isn’t a specific credence bigger than 1/2 that you know for sure you will have. We now need to do some tricksy stuff to overcome these two barriers to a counterexample to RP+.

The specific time barrier is actually pretty easy. Suppose that a continuous (i.e., not based on frames, but truly continuously recording—this may require other laws of physics than we have) video tape is being made of your front door. You aren’t yourself observing your front door. You are out of the country, and will return around 17:00. At that point, you will have no new information on whether CG showed up in the afternoon or before the afternoon. An associate will then play the tape back to you. The associate will begin playing the tape back strictly between 17:59:59 and 18:00:00, with the start of the playback so chosen that that exactly at 18:00:00, CG won’t have shown up in the playback. However, you don’t get to see the clock after your return, so you can’t get any information from noticing the exact time at which playback starts. Thus, exactly at 18:00:00 you won’t know that it is exactly 18:00:00. However, exactly at 18:00:00, your credence that CG came in the afternoon will be bigger than 1/2, because you will know that the tape has already been playing for a certain period of time and CG hasn’t shown up yet on the tape. Thus, you know ahead of time that exactly at 18:00:00 your credence in the afternoon hypothesis will be higher than 1/2.

But you don’t know how much higher it will be. Overcoming that requires a second trick. Suppose that your associate is guaranteed to start the tape playback a non-infinitesimal amount of time before 18:00:00. Then at 18:00:00 your credence in the afternoon hypothesis will be more than 1/2 + α for any infinitesimal α. By RP+, before the tape playback, your credence in the afternoon hypothesis should be at least 1/2 + α for every infinitesimal α. But this is absurd: it should be exactly 1/2.

So, we now have a full counterexample to RP+, assuming infinitesimal probabilities and the coherence of the CG setup (i.e., something like (1)). At exactly 18:00:00, with no irrationality, memory loss or the like involved (ignorance of what time it is not irrational nor a type of memory loss), you will have a credence at least 1/2 + α for some positive infinitesimal α, but right now your credence should be exactly 1/2.

Appendix: Here’s an argument that if RP is true, so is RP+. For simplicity, I will work with real-valued probabilities. Suppose all the qualifications of RP hold, and you now are sure that at t₁ your credence in p will be at least r. Let X be a real number uniformly randomly chosen between 0 and 1 independently of p and any evidence you will acquire by t₁. Let C_t(q) be your credence in q at t. Let u be the following proposition: X < r/C_t(p) and p is true. Then at t₁, your credence in u will be (r/C_t(p))C_t(p) = r (where we use the fact that r ≤ C_t(p)). Hence, by RP your credence now in u should be r. But since u is a conjunction of two propositions, one of them being p, your credence now in p should be at least r.

(One may rightly worry about difficulties in dropping the restriction that we are working with real-valued probabilities.)

The light and clap game

Suppose a light turns on at a uniformly chosen random time between 10 and 11 am, not including 10 am, and Alice wins a prize if she claps her hands exactly once after 10 am but before the light is on. Alice is capable of clapping or not clapping her hands instantaneously at any time, and at every time she knows whether the light is already on.

It seems that no matter when the light turns on, Alice could have clapped her hands before that, and hence if she does not clap, she can be rationally faulted.

But is there a strategy by which Alice is sure to win? Here is a reason to doubt it. Suppose there is such a strategy, and let C be the time at which Alice claps according to the strategy. Let L be the time at which the light turns on. Then we must have P(10<C<L) = 1: the strategy is sure to work. But let’s think about how C depends on L. If L ≤ 10 + x, for some specific x > 0, then it’s guaranteed that C < 10 + x. But because the only information available for deciding at a time t is whether the light is on or off, the probability that we have C < 10 + x cannot depend on what exact value L has as long as that value is at least 10 + x. You can’t retroactively affect the probability of C being before 10 + x once 10 + x comes around. Thus, P(C<10+x|L∈[t,t+δ]) will be the same for any t ≥ 10 + x and any δ > 0. But P(C<10+x|L∈[10+x,11]) = 1. So, P(C<10+x|L∈[t,t+δ]) = 1 whenever t ≥ 10 + x and x > 0. By countable additivity, it follows that P(C≤10|L∈[t,t+δ]) = 1, which is impossible since C > 10. Contradiction!

So there is no measurable random variable C that yields the time at which Alice wins and that depends only on the information available to Alice at the relevant time. So there is no winning strategy. Yet there is always a way to win!

I don’t know how paradoxical this is. But if you think it’s paradoxical, then I guess it’s another argument for causal finitism.

Brownian motion and regret

Let B_t be a one-dimensional Brownian motion, i.e., Wiener process, with B₀ = 0. Let’s say that time 0 you are offered, for free, a game where your payoff at time 1 will be B₁. Since the expected value of a Brownian motion at any future time equals its current value, this game has zero value, so you are indifferent and go for it.

But here is a fun fact. With probability one, at infinitely many times t between 0 and 1 we will have B_t < 0 (this follows from Th. 27.24 here). At any such time, your expectation of your payout at B₁ to be negative. Thus, at infinitely many times you will regret your decision to play the game.

Of course, by symmetry, with probability one, at infinitely many times between 0 and 1 we will have B_t > 0. Thus if you refuse to play, then at infinitely many times you will regret your decision not to play the game.

So we have a case where regret is basically inevitable.

That said, the story only works if causal finitism is false. So if one is convinced (I am not) that regret should always be avoidable, we have some evidence for causal finitism.

Divine temporalism once again

I’m thinking about my recent argument against divine temporalism, the idea that God has no timeless existence but is instead in time, and time extends infinitely pastwards.

Here’s perhaps a simple way to make my argument go (I am grateful to Dean Zimmerman for suggestions that helped in this reformulation). If infinite time is a central feature of reality, as the temporalist says, then one of the most fundamental things for God to decide about the structure of creation is which of these three is to be true:

1. Nothing gets created.

2. There is creation going infinitely far back in time.

3. There is creation but it doesn’t go infinitely far back in time.

But without backwards causation, a temporal God cannot decide between (2) and (3). For at any given time, it’s already settled whether (2) or (3) is the case.

Now, it seems that the temporalist’s best answer is to deny the possibility of (2). We don’t expect God to choose whether to create square circles, and so if we deny the possibility of (2), God only needs to choose between (1) and (3).

But there are two issues with that. First, creation going infinitely far back in time is the temporalist’s best answer to the Augustinian question of why God waited as long as he did before creating—on this answer (admittedly contrary to Christian doctrine), God didn’t wait.

Second, and perhaps more seriously, there is the question of justifying the claim that (2) is impossible. There are four reasons in the literature for thinking that in fact creation has a finite past:

1. Big Bang cosmology

2. Arguments against actual infinity

3. Arguments against traversing an actually infinite time

4. Causal finitism.

None of these allow the temporalist to justify the impossibility of creation going infinitely far back in time. Big Bang cosmology is contingent, and does not establish impossibility. And if the arguments (ii) and (iii)
are good reasons for rejecting an infinite past of creation, they are also good reasons for rejecting divine temporalism, since divine temporalism would require God to have lived through an actually infinite time. And (iv) also seems to rule out divine temporalism. For suppose that in fact creation follows an infinite number of days without creation. During that infinite number of days without creation, on any day we could ask why nothing exists. And the answer is that God didn’t decide to create anything. So the emptiness of the empty day causally depends on God’s infinitely many decisions in days past not to start creating yet, contrary to causal finitism.

Domination and uniform spinners

About a decade ago, I offered a counterexample to the following domination principle:

1. Given two wagers A and B, if in every state B is at least as good as A and in at least one state B is better than A, then one should choose B over A.

But perhaps (1) is not so compelling anyway. For it might be that it’s reasonable to completely ignore zero probability outcomes. If a uniform spinner is spun, and on A you get a dollar as long as the spinner doesn’t land at 90^∘ and on B you get a dollar no matter what, then (1) requires you to go for B, but it doesn’t seem crazy to say “It’s almost surely not going to land at 90^∘, so I’ll be indifferent between A and B.”

But now consider the following domination principle:

2. Given two wagers A and B, if in every state B is better than A, then one should choose B over A.

This seems way more reasonable. But here is a potential counterexaple. Consider a spinner which uniformly selects a point on the circumference of a circle. Assume x is any irrational number. Consider a function u such that u(z) is a real number for any z on the circumference of the circle. Imagine two wagers:

• A: After the spinner is spun and lands at z, you get u(z) units of utility

• B: After the spinner is spun, the spinner is moved exactly x degrees counterclockwise to yield a new landing point z′, and you get u(z′) units of utility.

Intuitively, it seems absurd to think that B could be preferable to A. But it turns out that given the Axiom of Choice, we can define a function u such that:

3. For any z on the circumference of the circle, if z′ is the result of rotating z by x degrees counterclockwise around the circle, then u(z′) > u(z).

And then if we take the states to be the initial landing points of the spinner, B always pays strictly better than A, and so by the domination principle (2), we should (seemingly absurdly) choose B.

Remarks:

• The proof of the existence of u requires the Axiom of Choice for collections of countable sets of reals). In my Infinity book, I argued that this version of the Axiom of Choice is true. However, arguments similar to those in the book’s Axiom of Choice chapter suggest that the causal finitist has a good way out of the paradox by denying the implementability of the function u.

• Some people don’t like unbounded utilities. But we can make sure that u is bounded if we want (if the original function u is not bounded, then replace u(z) by arctan u(z)).

• Of course the function u is Lebesgue non-measurable. To see this, replacing u by its arctangent if necessary, we may assume u is bounded. If u were measurable and bounded, it would be integrable, and its Lebesgue integral around the circle would be rotation invariant, which is impossible given (3).

It remains to prove the existence of u. Let ∼ be the relation for points on the (circumference of the circle) defined by z ∼ z′ if the angle between z and z′ is an integer multiple of x degrees. This is an equivalence relation, and hence it partitions the circle into equivalence classes. Let A be a choice set that contains exactly one element from each of the equivalence classes. For any z on the circle, let z₀ be the point in A such that z₀ ∼ z. Let u(z) be the (unique!) integer n such that rotating z₀ counterclockwise around the circle by an angle of nx degrees yields z. Then for any z, if z′ is the result of rotating z′ by x degrees around the circle, then u(z′) = u(z) + 1 > u(z) and so we have (3).

A cosmological argument from the Hume-Edwards Principle

The Hume-Edwards Principle (HEP) says:

1. If you’ve explained every item in a collection, you’ve explained the whole collection of items.

This sounds very plausible, but powerful counterexamples have been given. For instance, suppose that exactly at noon, cannonball is shot out of a cannon. The collection C of cannonball states after noon has the property that each state in C is explained by an earlier state in C (e.g., a state at 12:01:00 is explained by a state at 12:00:30). By the Hume-Edwards Principle, this would imply that C is self-explanatory. But it plainly is not: it requires the cannon being fired at noon to be explained.

But I just realized something. All of the effective counterexamples to the Hume-Edwards Principle involve either circular causation or infinite causal regresses. We can now argue:

2. HEP is necessarily true.

3. If circular causation is possible, counterexamples to HEP are possible.

4. If infinite causal regresses are possible, counterexamples to HEP are possible.

5. So, neither circular causation nor infinite causal regresses are possible.

6. If there is no first cause, there is a causal circle or an infinite causal regress.

7. So, there is a first cause.

Similarly, it is very plausible that if infinite causal regresses are impossible, then causal finitism, the thesis that nothing can have an infinite causal history, is true. So, we get an argument from HEP to causal finitism.

Dialectically, the above is very odd indeed. HEP was used by Hume and Edwards to oppose cosmological arguments. But the above turns the tables on Hume and Edwards!

Objection: Not every instance of causal regress yields a counterexample to HEP. So it could be that HEP is true, but some causal regresses are still possible.

Response: It’s hard to see how there is sufficient structural difference between the cannonball story and other regresses to allow one to deny the cannonball story, and its relatives, while allowing the kind of regresses that are involved in Hume’s response to cosmological arguments.

Final remark: What led me to the above line of thought was reflecting on scenarios like the following. Imagine a lamp with a terrible user interface: you need to press the button infinitely many times to turn the lamp on, and once you do, it stays on despite further presses. Suppose now that in an infinite past, Alice was pressing the button once a day. Then the lamp was always on. Now I find myself with two intuitions. On the one hand, it seems to me that there is no explanation in the story as to why the lamp was always on: “It’s always been like that” just isn’t an explanation. On the other hand, we have a perfectly good explanation why the lamp was on n days ago: because it was on n + 1 days ago, and another button press doesn’t turn it off. And I found the second intuition pushing back against the first one, because if every day’s light-on state has an explanation, then there should be an explanation of why the lamp was always on. And then I realized this intuition was based on somehow finding HEP plausible—despite having argued against HEP over much of my philosophical career. And then I realized that one could reconcile HEP with these arguments by embracing causal finitism.

Shuffling infinitely many cards

Imagine there is an infinite stack of cards labeled with the natural numbers (so each card has a different number, and every natural number is the number of some card). In the year 2021 − n, you perfectly shuffled the bottom n cards in the stack.

Now you draw the bottom card from the deck. Whatever card you see, you are nearly certain that the next card will have a bigger number. Why? Well, let’s say that the card you drew has the number N on it. Next consider the next M cards in the deck for some number M much bigger than N. At most N − 1 of these have numbers smaller than N on them. Since these bottom M cards were perfectly shuffled during the year 2021 − (M + 1), the probability that the number you draw is bigger than N is at most (N − 1)/M. And since M can be made arbitrarily large, it follows that the probability that the number you draw is bigger than N is infinitesimal. And the same reasoning applies to the next card and so on. Thus, after each card you draw, you are nearly certain that the next card will have a bigger number.

And, yet, here’s something you can be pretty confident of: The bottom 100 cards are not in ascending order, since they got perfectly shuffled in 1921, and after that you’ve shuffled smaller subsets of the bottom 100 cards, which would not make the bottom 100 cards any less than perfectly shuffled. So you can be quite confident that your reasoning in the previous paragraph will fail. Indeed, intuitively, you expect it to fail about half the time. And yet you can’t rationally resist engaging in this reasoning!

The best explanation of what went wrong is, I think, causal finitism: you cannot have a causal process that has infinitely many causal antecedents.

A tension in some of my recent work

Here is a tension in some recent work of mine. In Chapter 6 of Infinity, Causation, and Paradox, I argue that (a) the Axiom of Choice for countable sets of reals (ACCR) is true, and (b) this version of the Axiom of Choice plus causal infinitism implies a nasty paradox, so we should accept causal finitism instead. The argument for ACCR makes use of the premise that mathematical entities exist necessarily. But in “Might All Infinities Be The Same Size?”, I argue that for all we know, some mathematical entities exist contingently. Thus, the latter paper undercuts the argument of Chapter 6 of the book.

Fortunately, the argument of Chapter 6 of the book looks like it might be fixable. The argument for ACCR proceeded as follows:

1. For any set of non-empty countable sets of reals, it is metaphysically possible that there is a choice function.

2. If possibly there is a choice function, then necessarily there is a choice function.

The argument for (1) is elaborate, but it is (2) that the considerations in my article block.

But we can try to proceed as follows. The paradox in Chapter 6 of the book requires a choice function for a particular collection of non-empty countable sets of reals (reals generatable by a certain infinitary coin-tossing process). By (1), there is a possible world w′ where that particular collection of sets does have a choice function. So it seems all we need to do is to run the paradox in w′, and we should be done.

There are probably other areas in Chapter 6 where some tweaking (or more than that!) is needed to make things work with mathematical contingentism, and hence my cautious wording.

If materialism is true, God exists

Causal finitism is the doctrine that backwards infinite causal histories are impossible.

1. If the xs compose y, then y cannot have caused all of the xs.

2. If materialism is true and causal finitism is false, then it is possible to have a human being that (a) is composed of cells and (b) caused each of its cells via a backwards infinite regress.

3. So, if materialism is true, causal finitism is true. (1, 2)

4. If causal finitism is true, then God exists.

5. So, if materialism is true, God exists. (3, 4)

6. If God exists, the materialism is false.

7. So, materialism is false. (5, 6)

Premise (1) is a strengthening of a plausible principle banning self-causation.

Premise (2) follows from the fact that we are causes of all our present cells. If presentism is true, that completes the argument against materialism as in my previous post. But if eternalism or growing block are true, then we may also be composed of our past cells. And we didn’t cause our first cells. However if causal finitism is false, then it’s very plausible that backwards infinite causal regresses are possible, and so we could have existed from eternity, continually producing new cells, with the old ones dying.

Premise (4) is backed by a version of the kalaam argument.

Premise (6) is definitional if we understand materialism strongly enough to apply not just to us but to all reality. If we understand materialism more weakly, then the argument “only” yields the conclusion (5) that if materialism is true, God exists.

The paradox of the Jolly Givers

Consider the Grim Reaper (GR) paradox. Fred’s alive at midnight. Only a GR can kill him. Each GR has an alarm with a wakeup time. When the alarm goes off, the GR looks to see if Fred’s alive, and if he is, the GR kills him. Otherwise, the GR does nothing. Suppose the alarm times of the GR’s are 12:30 am, 12:15 am, 12:07.5 am, …. Then Fred’s got to be dead, but no GR could have killed him. If, say, the 12:15 GR killed him, that means Fred was alive at 12:07.5, which means the 12:07.5 GR would have killed him.

A Hawthorne answer to the GR paradox is that the GRs together killed Fred, though no one of them did.

Here’s a simple variant that shows this can’t be true. You hang up a stocking at midnight. There is an infinite sequence of Jolly Givers, each with a different name, and each of which has exactly one orange. There are no other oranges in the world, nor anything that would make an orange. When a JG’s alarm goes off, it checks if there is anything in the stocking. If there is, it does nothing. If there is nothing in the stocking, it puts its orange in the stocking. The alarm times are the same as in the previous story.

The analogous Hawthorne answer would have to be that the JGs together put an orange in the stocking. But then one of the JGs would need to be missing his orange. But no one of the JGs is missing his orange, since no one of them took it out of his pocket. So, the orange would have had to come out of nowhere.

And, to paraphrase a very clever recent comment, if it came out of nowhere, why would it be an orange, rather than, say, a pear?

I think the JG paradox also suggests an interesting link between the principle that nothing comes from nothing and the rejection of supertasks.

Causal finitism and functionalism

Say that a possible thought content has finite complexity provided that the thought content can be represented by a sentence of finite length in a language whose basic terms are the fundamental concepts in the thought content.

1. Necessarily, if functionalism is true, then the occurrence of a thought content with infinite complexity requires infinitely many states to cooperate to produce a single effect.

2. Infinitely many states cannot cooperate to produce a single effect.

3. It is possible for a thought content with infinite complexity to occur.

4. So, functionalism is false.

I have two separate ideas to defend (1). First, it seems like a system capable of producing a thought content must go through a number of states proportional to the complexity of that thought content in producing it if functionalism is true. Second, the occurrence of a thought content of infinite complexity requires infinitely many constituent states. Moreover, thoughts have to be unified: to think the conjunction of p and q is not just to think p and to think q but to think them in a unified way. On functionalism, the unification has to be causal in nature. To unify the infinitely many constituent states would require them to have the capability of producing some effect together.

If I were a functionalist, I would deny (3). The cost of that is that then most truths end up unthinkable, which seems implausible.

Leaving space

Suppose that we are in an infinite Euclidean space, and that a rocket accelerates in such a way that in the first 30 minutes its speed doubles, in the next 15 minutes it doubles again, in the next 7.5 minutes it doubles, and so on. Then in each of the first 30 minutes, and the next 15 minutes, and the next 7.5 minutes, and so on, it travels roughly the same distance, and over the next hour it will have traveled an infinite distance. So where will it be? (This is a less compelling version of a paradox Josh Rasmussen once sent me. But it’s this version that interests me in this post.)

The causal finitist solution is that the story is impossible, for the final state of the rocket depends on infinitely many accelerations, and nothing can causally depend on infinitely many things.

But there is another curious solution that I’ve never heard applied to questions like this: after an hour, the rocket will be nowhere. It will exist, but it won’t be spatially related to anything outside of itself.

Would there be a spatial relationship between the parts of the rocket? That depends on whether the internal relationships between the parts of the rocket are dependent on global space, or can be maintained in a kind of “internal space”. One possibility is that all of the rocket’s particles would lose their spatiality and exist aspatially. Another is that they would maintain spatial relationships with each other, without any spatial relationships to things outside of the rocket.

While I embrace the causal finitist solution, it seems to me that the aspatial solution is pretty good. A lot of people have the intuition that material objects cannot continue to exist without being in space. I don’t see why not. One might, of course, think that spatiality is definitive of materiality. But why couldn’t a material object then continue to exist after having lost its materiality?

Thomson's core memory paradox

This is a minor twist on the previous post.

Magnetic core memory (long obsolete!) stored bits in the magnetization of tiny little rings. It was easy to write data to core memory: there were coils around the ring that let you magnetize it in one of two directions, and one direction corresponded to 0 and the other to 1. But reading was harder. To read a memory bit, you wrote a bit to a location and sensed an electromagnetic fluctation. If there was a fluctuation, then it follows that the bit you wrote changed the data in that location, and hence the data in that location was different from the bit you wrote to it; if there was no fluctuation, the bit you wrote was the same as the bit that was already there.

The problem is that half the time reading the data destroys the original bit of data. In those cases—or one might just do it all the time—you need to write back the original bit after reading.

Now, imagine an idealized core not subject to the usual physics limitations of how long it takes to read and write it. My particular system reads data by writing a 1 to the core, checking for a fluctuation to determine what the original datum was, and writing back that original datum.

Let’s also suppose that the initial read process has a 30 second delay between the initial write of the 1 to the core and the writing back of the original bit. But the reading system gets better at what it’s doing (maybe the reading and writing is done by a superpigeon that gets faster and faster as it practices), and so each time it runs, it’s four times as fast.

Very well. Now suppose that before 10:00:00, the core has a 0 encoded in it. And read processes are triggered at 10:00:00, 10:00:45, 10:00:56.25, and so on. Thus, the nth read process is triggered 60/4ⁿ seconds before 10:01:00. This process involves the writing of a 1 to the core at the beginning of the process and a writing back of the original value—which will always be a 0—at the end.

Intuitively:

1. As long as the memory is idealized to avoid wear and tear, any possible number—finite or infinite—of read processes leaves the memory unaffected.

By (1), we conclude:

2. After 10:01:00, the core encodes a 0.

But here’s how this looks from the point of view of the core. Prior to 10:00:00, a 0 is encoded in the core. Then at 10:00:00, a 1 is written to it. Then at 10:00:30, a 0 is written back. Then at 10:00:45, a 1 is written to it. Then at 10:00:52.5, a 0 is written back. And so on. In other words, from the point of view of the core, we have a Thomson’s Lamp.

This is already a problem. For we have an argument as to what the outcome of a Thomson’s Lamp process is, and we shouldn’t have one, since either outcome should be as likely.

But let’s make the problem worse. There is a second piece of core memory. This piece of core has a reading system that involves writing a 0 to the core, checking for a fluctuation, and then writing back the original value. Once again, the reading system gets better with practice. And the second piece of core memory is initialized with a 1. So, it starts with 1, then 0 is written, then 1 is written back, and so on. Again, by premise (1):

3. After the end of the reading processes, we have a 1 in the core.

But now we can synchronize the reading processes for the second core in such a way that the first reading occurs at 9:59:30, and space out and time the readings in such a way that prior to 9:59:30, a 1 is encoded in the core. At 9:59:30, a 0 is written to the core. At 10:00:00, a 1 is written back to the core, thereby completing the first read process. At 10:00:30, a 0 is written to the core. At 10:00:45, a 1 is written back, thereby completing a second read process. And so on.

Notice that from around 10:00:01 until, but not including, 10:01:00, the two cores are always in the same state, and the same things are done to it: zeroes and ones are written to the cores at exactly the same time. But when, then, do the two cores end up in different final states? Does the first core somehow know that when, say, at 10:00:30, the zero is written into it, that zero is a restoration of the value that should be there, so that at the end of the whole process the core is supposed to have a zero in it?

Another way to turn Thomson's Lamp into a real paradox

In Thomson’s Lamp, a lamp is (say) off at 10:00, and the switch is toggled at 10:30, 10:45, 10:52.5, and so on, and we are asked whether the lamp is on or off at 11:00, neither option being satisfactory.

As it stands, Thomson’s Lamp is a puzzle rather than a paradox. There does not seem to be any absurdity in the answer being “on” or the answer being “off”.

In Infinity, Causation and Paradox I tried to generate a paradox from Thomson’s Lamp. But here is perhaps a better way. Start with this premise:

1. Removing any number of interactions with a system none of which changes a system will not affect the system.

Now, consider these complex interactions with the lamp system:

• Toggling the switch at 10:30 and at 10:45

• Toggling the switch at 10:52.5 and at 10:56.25

• …

Two successive togglings do nothing, so each of these is an interaction that does nothing. By 1, removing them all makes no difference. Now, we know that if we remove them all, the lamp will be off at 11:00, since its switch will not have been toggled even once since 10:00. So, we have established:

2. The lamp will be off at 11:00.

But now consider these complex interactions:

• Toggling the switch at 10:45 and at 10:52.5

• Toggling the switch at 10:56.25 and at 10:58.125

• …

Again, each of these is an interaction that makes no difference. So if we remove them all, by 1 that won’t change anything. But if we remove all these interactions, we have a lamp that is on at 10:31 (since we still have the 10:30 toggling) and then never has its switch toggled. Thus, we have shown:

3. The lamp will be on at 11:00.

So, indeed, we now have a paradox.

Continuity of time and causation

In a standard causal deterministic system, given three times t₁ < t₂ < t₃, the state of the system at t₁ causes the state of the system at t₃ by means of the state of the state of system at t₂. If time is infinitely subdivided, then the state of the system at t₂ causes the state of the system at t₃ by means of the state of the system at t_2.5 (where t₂ < t_2.5 < t₃), and so on. This is an infinite regress. And it’s vicious, because it’s a dependency regress.

Here is one way to see that it’s a dependency regress. Imagine a really unpleasant situation where you need to kill Hitler, but the only way to kill Hitler is to initiate a continuous causal chain that proceeds through Hitler’s uncountably many henchmen, set up so that at every time strictly between t₁ and t₂ a henchman dies, and their death is caused by the death of each previously dead henchman; at t₁, Himmler is directly shot by you, and at t₂, Hitler dies because of the previous henchman deaths. It is clear that in this case every henchman’s death is intended as a means to Hitler’s death. This matters morally. If it turns out that any person in the causal chain is actually innocent, then the Principle of Double Effect will not allow you to kill Hitler by killing Himmler. For the causal chain from Himmler’s death through to Hitler’s proceeds by means of that innocent. But an outcome depends on its means, so a regress of means is a dependency regress.

If there are no vicious infinite regresses, it follows that one cannot have deterministic causal chains intimately tied to infinitely subdivided time. In fact, I think nothing hangs on determinism here.