A new argument for causal finitism

I will give an argument for causal finitism from a premise I don’t accept:

1. Necessary Arithmetical Alethic Incompleteness (NAAI): Necessarily, there is an arithmetical sentence that is neither true nor false.

While I don’t accept NAAI, some thinkers (e.g., likely all intuitionists) accept it.

Here’s the argument:

2. If infinite causal histories are possible, supertasks are possible.

3. If supertasks are possible, for every arithmetical sentence, there is a possible world where someone knows whether the sentence is true or false by means of a supertask.

4. If for every arithmetical sentence there is a possible world where someone knows whether the sentence is true or false by means of a supertask, there is a possible world where for every arithmetical sentence someone knows whether it is true or false.

5. Necessarily, if someone knows whether p is true or false, then p is true or false.

6. So, if infinite causal histories are possibly, possibly all arithmetical sentences are true or false. (2-5)

7. So, infinite causal histories are impossible. (1, 6)

The thought behind (3) is that if for every n it is possible to check the truth value of ϕ(n) by a finite task or supertask, then by an iterated supertask it is possible to check the truth values of ∀xϕ(x) (and equivalently ∃xϕ(x)). Since every arithmetical sequence can be written in the form Q₁x₁...Q_kx_kϕ(x₁,...,x_k), where the truth value of ϕ(n₁,...,n_k) is finitely checkable, it follows that every arithmetical sequence can have its truth value checked by a supertask.

The thought behind (4) is that one can imagine an infinite world (say, a multiverse) where for every arithmetical sentence ϕ the relevant supertask is run and hence the truth value of the sentence is known.

A supertasked Sleeping Beauty

One of the unattractive ingredients of the Sleeping Beauty problem is that Beauty gets memory wipes. One might think that normal probabilistic reasoning presupposes no loss of evidence, and weird things happen when evidence is lost. In particular, thirding in Sleeping Beauty is supposed to be a counterexample to Van Fraassen’s reflection principle, that if you know for sure you will have a rational credence of p, you should already have one. But that principle only applies to rational credences, and it has been claimed that forgetting makes one not be rational.

Anyway, it occurred to me that a causal infinitist can manufacture something like a version of Sleeping Beauty with no loss of evidence.

Suppose that:

• On heads, Beauty is woken up at 8 + 1/n hours for n = 2, 4, 6, ... (i.e., at 8.5 hours or 8:30, at 8.25 hours or 8:15, at 8.66… hours or 8:10, and so on).

• On tails, Beauty is woken up at 8 + 1/n hours for n = 1, 2, 3, ... (i.e. at 9:00, 8:30, 8:20, 8:15, 8:10, …).

Each time Beauty is woken up, she remembers infinitely many wakeups. There is no forgetting. Intuitively she has twice as many wakeups on tails, which would suggest that the probability of heads is 1/3. If so, we have a counterexample to the reflection principle with no loss of memory.

Alas, though, the “twice as many” intuition is fishy, given that both infinities have the same cardinality. So we’ve traded the forgetting problem for an infinity problem.

Still, there may be a way of avoiding the infinity problem. Suppose a second independent fair coin is tossed. We then proceed as follows:

• On heads+heads, Beauty is woken up at 8 + 1/n hours for n = 2, 4, 6, ...

• On heads+tails, Beauty is woken up at 8 + 1/n hours for n = 1, 3, 5, ...

• On tails+whatever, Beauty is woken up at 8 + 1/n hours for n = 1, 2, 3, ....

Then when Beauty wakes up, she can engage in standard Bayesian reasoning. She can stipulatively rigidly define t₁ to be the current time. Then the probability of her waking up at t₁ if the first coin is heads is 1/2, and the probability of her waking up at t₁ if the first coin is tails is 1. And so by Bayes, it seems her credence in heads should be 1/3.

There is now neither forgetting nor fishy infinity stuff.

That said, one can specify that the reflection principle only applies if one can be sure ahead of time that one will at a specific time have a specific rational credence. I think one can do some further modifying of the above cases to handle that (e.g., one can maybe use time-dilation to set up a case where in one reference frame the wakeups for heads+heads are at different times from the wakeups for heads+tails, but in another frame they are the same).

All that said, the above stories all involve a supertask, so they require causal infinitism, which I reject.

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.

The four causes and supertasks

Suppose I make a geologist’s hammer out of a chunk of steel and break a rock with the hammer. Then the chunk of steel is the material cause of the hammer, and the hammer is the efficient cause of the rock breaking.

The hammer then is explanatorily prior to the rock breaking and the chunk of steel is explanatorily prior to the hammer.

Admittedly these are different kinds of explanatory priority. But they do nonetheless combine: it is clearly correct to say that the chunk of steel is explanatorily prior to the rock breaking. (I am not claiming that transitivity holds across all the kinds of explanatory priority, though I suspect it does, but only here.) But now notice that this instance of explanatory priority does not correspond to any of the four causes: in particular the chunk of steel is neither the material nor the efficient cause of the rock breaking (it is only insofar as the chunk was shaped into a geologist’s hammer that it broke the rock). Hence, the four causes do not exhaust all the types of explanatory priority.

Other examples are possible. I push a rock with my hand, and consider the conjunctive state HM of there being a hammer and a rock moving. Then HM is explained by the chunk of steel and my hand. But the chunk of steel and my hand constitute neither a material or not an efficient (nor any other) cause of HM. Thus, again, we have explanatory priority not corresponding to one of the four causes.

The above examples do, however, permit one to hold the following view:

1. All fundamental instances of explanatory priority are instances of the four causes.

Thus, the four causes would be like Aristotle’s four elements or three types of friendship: they combine to provide all the cases.

But now an interesting bit of heavy-duty metaphysics. Suppose that dense causal sequences are possible, i.e., causal sequences such that between any two items in the sequence there is an intermediate one. Then no instance of causation in the dense sequence will be fundamental. And hence (1) won’t tell us as much as it seems to. Indeed, given dense causal sequences, weakening the four cause thesis to (1) eviscerates the four cause thesis.

Thus we have an argument that if we want to take the four cause thesis seriously, we need to accept (1), and hence we need to reject dense causal sequences.

But if supertasks are possible, it seems like dense causal sequences should be possible. So, if we want to take the four cause thesis seriously, we need to reject supertasks.

It is, by the way, interesting to think about supertasks where the items in the task alternate between different types of causation.

Note that the above point applies to other sparse pluralisms about causation besides the four-cause one.

Recognizing the finite

We have a simple procedure for recognizing finite sequences. We start at the beginning and go through the sequence one item at a time (e.g., by scanning with our eyes). If we reach the end, we are confident the sequence was finite. This procedure can be relied on if and only if there are no supertasks—i.e., if and only if it is impossible to have an infinite sequence of tasks started and completed.

How do we know that there are no supertasks? Either empirically or a priori. To know it empirically, we would have to know that the various tasks we’ve completed were finite. But how would we know of any tasks we’ve completed that it’s finite if not by the above procedure?

So we have to know it a priori.

And the only story I know of how we could do that is by a priori cognizing some anti-infinity principle like Causal Finitism.

I am not sure how strong the above argument is. It is a little too close to standard sceptical worries for comfort.

Thomson's lamp and two counterfactauls

Thomson's lamp toggles each time you press the button and nothing else affects its state. The lamp is on at noon, and then a supertask consisting of infinitely many button presses that completes by 1 pm, and the question is whether the light is on or off at 1 pm. There is no contradiction yet. But now add these two claims:

1. The state of the lamp at 1 pm would not be affected by shifting the times at which the button presses happen, if (a) all the button presses happen between noon and 1 pm, and (b) we ensure that no two button presses happen simultaneously.
2. If we removed one button press from the sequence of button presses between noon and 1 pm, the state of the lamp at 1 pm would not change.

Given this intuition, we do have a problem. Suppose that our sequence of supertask button presses occurs at 12:30, 12:45, 12:52.5, and so on. Then shift this sequence of button presses forward in time, so that now the sequence is at 12:45, 12:52.5, 12:45.25,and so on. By (1) this wouldn't affect the outcome, but by (2) it would as we will have gotten rid of the first button press. That's a contradiction.

So if we think Thomson's lamp is possible--which I do not--we need to deny at least one of the two counterfactuals. I think the best move would be simply to deny both (1) and (2), on the grounds that the connection between the state of the lamp at 1 pm and the button presses must be indeterministic.

Thomson's lamp and the Axiom of Choice

Consider Thomson's lamp: a lamp with a pushbutton switch that toggles it on and off. The lamp starts in the off position, and then in the next half minute the button is pressed, and in the next quarter it is pressed again, and then in the neight eighth again, and so on. Then at the end of the supertask, the lamp is either on or off.

Now keep the lamp but change the story. During each of the ever shorter intervals, a coin is flipped and the switch is pressed if it lands heads, and not pressed if it lands tails. Moreover, the final state of the lamp depends on the results of the coin flips in the following ways:

1. The results of the coin flips determine the final state of the lamp.
2. For any sequence of coin flip results, if any one (and only one) coin flip had a different, the lamp's final state would have been different, too.

Surprisingly, the existence of a lamp that would work in this way implies a version of the Axiom of Choice. To see this, notice that if the coin flips are independent and fair, then the subset of the probability space where the lamp's final state is on is nonmeasurable.[note 1]

But of course, on some technical assumptions, the existence of a nonmeasurable set requires a version of the Axiom of Choice. So if we read the Thomson's lamp story in such a way that the final outcome is determined by which presses are made and which aren't, in such a way that changing a single press changes the final outcome, that story seems to commit us to a version of the Axiom of Choice.

Conversely, it is easy to use the Axiom of Choice for pairs to prove the existence of a function such as would be implemented by the lamp.[note 2]