Sunday, June 23, 2013

Cannonball and regress

From my first year of graduate school, I've been pushing a cannonball argument against Hume's idea that a regress of causes provides a complete explanation. The argument isn't very complex, but it has some complexity (it talks of complete states and all that), and it has occurred to me that it can be simplified.

At noon the cannonball is at rest and precisely then a cannon is fired. At every time after noon, the cannonball is moving. (Maybe the whole thing takes place in space.) We now have this dialogue:

  • You: Why is the cannonball ball moving at one minute after noon?
  • Me: Because it's moving at half a minute after noon and there is inertia.
  • You: But why is the cannonball ball moving at half a minute after noon?
  • Me: Because it's moving at a quarter of a minute after noon and there is inertia.
  • You: But why is the cannonball ball moving at a quarter of a minute after noon?
  • Me: Because it's moving at an eighth of a minute after noon and there is inertia. Don't you see the pattern?
  • You: I do see the pattern, but why is it moving at any of these times: a minute after noon, half a minute after noon, a quarter of a minute after noon, an eighth of a minute after noon and so on? Why is the cannonball moving at all at any time after noon?
  • Me: Don't you see, I've explained each item in the chain, and so I've explained the chain!
You should object that as long as I haven't mentioned the cannon being fired, I haven't explained why it is that the cannonball is moving at all. If my chainwise explanation was a good explanation, then a complete explanation of all the motion of the of the cannonball could be given without mentioning the cannon. And that's absurd. So an infinite chain does not give an explanation of itself.

Now suppose that there was no such time as noon and no cannon. Would the infinite chain then explain itself? No! For the chain is no more explanatory if there is
no such time as noon and no cannon. Taking away the real explanation does not turn the chain into an explanation.

16 comments:

Hassan Abdillah said...

Dr. Pruss,

Do forgive me if these questions sound overly simplistic!

I'm a bit confused whether the extrapolation of the analogy's conclusion to a noon- and cannon-less scenario is justified. In this case, we seem to assume, even before making the argument, that there is an explanation (the cannon)- and hence if the explanation is removed, surely the (now explanation-less) chain wouldn't explain itself. So I think in this analogy, the assumption of a prior explanation has already been built in.

For example, if we assume that an explanation (say X) for an infinite chain of contingent events exist, and THEN we ask if the chain can explain itself without resorting to X, the answer will be obviously no. But what if we start with an infinite chain of contingent event without any assumption as regards its explanation (say we don't know if the cannonball has indeed been fired), would it be as obvious that it requires an explanation?

My second question here is regarding actual vs. potential infinity. You have to forgive my ignorance here, the only material on this issue I have read thus far is Dr. Craig's discussion on it in context of the KCA. An infinite chain of contingent things is actually infinite, but the time intervals between noon and a later point in time (during the cannonball's flight) is potentially infinite (right?). If this is true, wouldn't you say the analogy is misplaced? Because clearly, the cannon wasn't fired an infinite amount of time ago, just that the finite amount of time can be broken into an infinite number of intervals.

I don't see how supposing "there is no noon" answers this problem either: does it replace the potential infinite with an actual infinite (i.e. the cannon was fired an infinite time ago)?

Without the noon- and cannon- built in the scenario, what we have is no more than this: a cannonball has been moving from eternity past. Assuming that the cannon fired the cannonball a finite time ago, the cannonball's flight definitely calls out for an explanation, even considering the chain by itself. But I don't see how the proposition "a cannonball has been moving from eternity past" needs an explanation above and beyond itself.

Many thanks in advance
Hassan

Angra Mainyu said...

Hi, Alexander

I don't find the argument persuasive for a number of reasons, but for example, I would raise the following two objections:

Objection 1:

Let pt be the proposition that reports the condition of the ball at t (momentum, etc.), for any time t in the open interval between noon and one minute past noon. Let p be the conjunction of all of the pt -
Let q be the proposition that the ball changed from a state of rest to a state of movement

Then q and p are different propositions, and they call for different explanations. It's true that, intuitively – given what we know about cannons, etc. -, we reckon that if p is true, then so is q, so we're intuitively asking for an explanation of q: But in the case of an infinite past and no beginning, the scenario is very different, since there is no change from an initial state to another state.


Objection 2:

2.1. We're not justified in applying Newton's laws or any other present-day model to arbitrarily small amounts of time, so this is a case in which we do not have a [proper] explanation of every one of the individual conjuncts.
So, it seems the items in the chain are not all explained by previous items.
If so, then even leaving objection 1. aside, it seems that the cannonball example does not provide a counterexample against a proposed principle that states that if each of the conjuncts has been explained by one or more previous ones (or every item in a chain, etc.), then one has an explanation of the conjunction (or one such explanation may be built out of that), nor is it evidence against the weaker claim that at least sometimes one has such an explanation.

2.2. Let's leave point 1. aside, and let's assume for the sake of the argument that the universe is actually such that there are laws that we can correctly express with some formulas F that properly apply to any arbitrarily short period, and that we know the formulas in question. Let's also leave aside issues such as whether we should talk of particles rather than cannonballs, when considering such brief periods.

In that case, we have two possibilities:

2.2.a. The process of the cannon's firing took place entirely during some temporal interval I contained in some interval (t0, noon], where t0 is some time long before the cannon was fired. If so, then ball is not at rest at noon, since – for instance – the energy has already been transferred to the cannonball at noon, and part of it even before noon.
So, that contradicts the scenario.
Thus, it seems that according to the cannonball scenario:

2.2.b. The firing of the cannon continued at least for some time after noon, and thus, for t sufficiently small, the individual conjunct pt is not explained by a previous conjunct, plus the laws F and environmental conditions not including the firing of the cannon. Hence, it seems that the cannonball example does not provide a counterexample against a proposed principle that states that if each of the conjuncts has been explained by one or more previous ones (or every item in a chain, etc.), then one has an explanation of the conjunction (or one such explanation may be built out of that), nor is it evidence against the weaker claim that at least sometimes one has such an explanation.

Alexander R Pruss said...

Hassan:

The question here isn't whether it "requires" an explanation, but whether it is explained by the chain. Hume thinks the chain explains itself. That is surely wrong.

I also don't see why a backwards infinite amount of time makes for an explanation. Suppose the cannonball has always been moving. And I ask: Why has it always been moving? Surely the answer can't be: Because it always has been moving.

Angra:

ad 1: Sure, q is different from p, but I am not asking for an explanation of q. Actually, what I am asking for in this argument is an explanation for the disjunction of all the pt.

Suppose there was no change in the cannonball's state--maybe it came into existence already moving. Still, the point remains--the cannonball's state doesn't explain itself.

ad 2.1: I think we are justified in assuming that well-confirmed laws (though Newton's aren't such, I grant) apply in all domains. Scientific realism requires something like this.

But, yes, I am happy to run the argument in a counterfactual world where something like Newton's laws hold in full generality.

ad 2.2: You can fill out the details in one of two ways.

(i) The cannon is in a world where cannons have the power to instantaneously impart a velocity to cannonballs. So the cannonball has no velocity at noon and has large velocity at every time after noon.

(ii) The cannon continues to impart energy to the cannonball for a short amount of time after noon. But this does not affect the argument. For I am not asking here for an explanation of why the cannonball has this precise velocity at, say, 0.25 seconds past noon. I am simply asking for an explanation of why the cannonball has nonzero velocity at that time. And that is sufficiently explained by the fact that it had nonzero velocity earlier and there is inertia. If you want, you can add to the explanans that there was no force that counteracted that velocity (the push from the expanding gases in the cannon does not counteract the velocity). Of course, in this case there are *two* explanations of why it's moving at a given time: one in terms of inertia and an earlier velocity (which is sufficient) and another in terms of the gases pushing. But that doesn't affect the argument.

Angra Mainyu said...

Alex,

ad. 1.

Okay, but then you would have to argue that even if we have an explanation of every single one of the pt, we do not have an explanation of the conjunction of them – not just that we do not have an explanation as to why the ball began to move in the first place.

So far, I've not seen evidence that even though you have an explanation of each of the pt, you don't have an explanation of the conjunction of the pt (though admittedly, I've not seen the explanation of all of the pt).

Regarding the cannonball that came into existence already moving, I would say that the scenario requires more details. Are you saying that the cannonball does not exist at noon, but exists at any time after noon?

In that case (please let me know if I misunderstand), you would have to show that explaining each of the pt does not explain the conjunction of all of the pt (regardless of the issue of the explanation of the fact that the ball comes into existence). As before, I do not see evidence that works.

ad. 2.1.


2.1.a. This particular point is now a side note given the counterfactual world, but I would say that if we're justified in assuming that all well-confirmed laws apply in all domains, including any interval no matter how small, then I would say that there are no well-confirmed laws (though that is a result of the restrictions resulting from the criterion you propose, not my position), since we do not seem to be justified to apply the models we do have to such short periods, as far as I can tell (e.g., how do you account for gravity and the other forces together without a Quantum Gravity theory?).

2.1.b. Moreover, it might be that there are no such short periods. What if, say, time is discrete in an Aristotelian sense? (for instance). It seems that we need to also assume that time is dense, since the point of cannonball example (as I understand it; please let me know if I'm wrong) is to show an example in which we would have a [proper] explanation of each of the pt in terms of some pt' for t < t', plus some laws. But that requires temporal density.

In addition, it seems that that also requires that we know that time is dense, and that we know they may properly apply the formulas in question (say, F, whether they're Newton's laws or something else) to any interval, no matter how short.

Even then, it seems that we still wouldn't know the conditions at each t, anyway, so that might be another difficulty (i.e., it seems that we would still be assuming that we have such an explanation, even though we do not).

All that said, I think that we should switch from "we" to "they", where "they" are the people who live at that world, since their situation seems to be very different from ours (also considering the next point).

Angra Mainyu said...

ad 2.2.

2.2.(i).
Okay, so let's say that Castiel and Uriel live at world W1, where time is dense, they know that time is dense, they may properly apply formulas N to any temporal interval, etc., and there are cannons and cannonballs, etc.
Also, cannons have the property you stated. But how does that work?
One may argue as follows:
At any t before t0 (i.e., noon), the cannon is at rest is and so is the ball. So far, so good.
Now, exactly at t0, the ball remains at rest. So, it has no momentum at t0.
But that means that the cannon (or gases, etc.) has not transferred any momentum to the ball, by any means, in the interval (t1, t0] (where t1 is an earlier time). But that seems to imply that the cannon does all of the firing in an interval J included in the open interval (t0, t0+1] (1 is one minute), which entails that the explanation of at least some of the pt includes what the cannon does.

So, it seems to me that that still doesn't work; maybe you're thinking of a world. But I may have misunderstood what you meant; if so, please let me know.


2.2.(ii).

As far as I can tell, you seem to be asking for an explanation of the conjunction of all of the pt, not only for an explanation as to why the cannonball has non-zero velocity at t=t0+0.25, which may very well be explained in terms of what happened at t0+0.14 (for instance).
But for some pt, you don't have an explanation in terms of some previous pt' plus N, since you also need to state what the cannon is doing.

Moreover, if the ball is at rest at noon, it seems all of the action of the cannon took place after noon, so a proper explanation of each of the pt would require to state, for some of the t, what the cannon is doing. The conjunction of those explanations would seem to include everything that the cannon did.

skepticAl said...

I like it! It's simple and it works.

Hassan Abdillah said...

I see the point in Angra Manyu's above post, Dr. Pruss. In the cannonball analogy you presented, the explanation of the ball's motion at any time was restricted to the ball's state at an earlier time in conjunction of the relevant laws. I don't think this sufficiently explains the ball's motion, however- what the cannon was doing needs also be stated. So the "chain of events" in question would involve not only the ball, but also the cannon's firing. It seems intuitive to assume that this chain (ball's motion+cannon's firing) does indeed explain itself, without needing any external explanation.

Also, if the cannonball was always moving (from eternity, without any change of state whatsoever), then I don't see how it's obvious that it needs a cause over and above the chain itself. Would you kindly explain further?

Many thanks,
Hassan

Alexander R Pruss said...

Hassan:

Regarding the first issue, one can have explanatory overdetermination. Suppose you have a car that's moving at 80 km/h, while the driver is holding down the accelerator. Three seconds later, the car is still moving forward. Why? There are two different explanations one can give: (a) it was moving at 80 km/h and there was inertia, and (b) the accelerator was pushed.

Each of these is a sufficient explanation on its own.

Regarding the second, we should separate out the issue of whether a cause is "required" and the question of explanation. Right now we are discussing the question of explanation, and my claim is that the infinite chain is just as mysterious when it extends infinitely far back as when it extends a finite distance. In both cases you don't know why it is that the chain has items.

Angra Mainyu said...


Alex,

The explanation that it was moving at some t' prior than t, plus the laws would not explain why it was moving at t in the way it was moving. The explanation would not be sufficient.

But if you're trying to explain that it was still moving, that partial account might or might not explain it in their universe, what they know and what they're trying to find out. For example, if one asks why there was an explosion at a certain location, a potential explanation is that there was some C4 programmed to go off at that time. But that explanation might be entirely inadequate for the task at hand, if the person asking for an explanation is trying to figure out how or why the explosives got there in the first place.
Also, it may well be that the C4 explanation does not work if, say, the person asking for the explanation does not know what C4 is, or what it is to program something to go off (say, she's from a hunter-gatherers tribe with no previous contact with modern technology; surely, the C4 account won't explain the explosion to her)

I still do not see how you have an explanation of each of the pt, without having one of the conjunction of them in this case.

But we may consider a hypothetical dialogue in their universe:

Castiel: Why was the cannonball moving at one minute after noon?
Uriel: Because it's moving at half a minute after noon and there is inertia.
Castiel: Okay, that explains it. Thanks. Why was it moving half a minute after noon.
Uriel: Because it's moving at a quarter of a minute after noon and there is inertia.
Castiel: Okay, that explains it. Thanks. Why was it moving a quarter of a minute after noon?
Uriel: Because it's moving at an eighth of a minute after noon and there is inertia.
Castiel: Thanks, that explains why it's moving a quarter of a minute past noon. Also, I see a pattern. I see that by pointing to that pattern, you have explained to me, for each one single time t(n)=noon+1/(2^n) minutes, why the cannonball was moving at t(n).
Uriel: Hmm...I wasn't sure about that, but okay. Then, do you have any other questions?
Castiel: Yes, you haven't explained the infinite conjunction of the statements S(n):= 'The ball was moving at t(n)', for all n. I want an explanation of the conjunction S.
Uriel: I have to admit that I don't know I have explained each S(n). That would require me to explain infinitely many of them, and in none of them I included the actions of the cannon. However, if you consider the explanation of each of the S(n) to be sufficient, I'm not sure why you don't find the explanation of S sufficient.
Perhaps, what you want to know is why the ball began to move in the first place?
But then, you're asking for something else. In that case, I would have to include the actions of the cannon.
Castiel: So, you need to list what happened at noon, or before noon?
Uriel: I do not know that I have to, for the following reason: You see, at noon, the ball was at rest, and so the cannon had not yet transferred any momentum to the ball. So, the cannon did all the work later.
Castiel: Alright, so you can explain S without saying what happened at noon, or before, but you still cannot explain S with S. You need something else.
Uriel: But then, it seems you no longer accept the very partial and incomplete explanation of each of the S(n). Otherwise, why would you not accept them as explaining the conjunction?

I get you may disagree with some of the above. I would ask what that is.

Angra Mainyu said...

But how about a more detailed principle, in order to account for the fact that people may want or need different levels of explanation (more detailed, complete, etc.) in different contexts?

Let's say that P is some infinite class of propositions, and C is the conjunction of all of the P in the class – let's assume that said conjunction is coherent.
Let's further say that for each single proposition P in P Uriel has [somehow] provided an explanation in terms of other propositions in P that is sufficient to explain that specific P to Castiel to any degree Castiel may want an explanation of P, in any context.
In that case, it seems to me there is nothing else Castiel may want to ask Uriel, regarding how to explain C. Do you agree or disagree?

Alexander R Pruss said...

Regarding your second question, I disagree. Of course, I can't use my simplified argument in this case. Instead, I might suppose a deterministic physics, and let the propositions be complete descriptions of the state of the universe at times after noon.

But the more important point, perhaps, is that I am not working with conjunctions but with disjunctions. I am not asking a question akin to why all the elephants exist, but one akin to why there any elephants at all.

I think this may matter. Suppose we want to know why there are (tenseless) any elephants at all, and I say: Well, Dumbo is an elephant, and Dumbo's parents conceived him. Not only this doesn't explain why there are any elephants at all, but because the explanation presupposes two elephants, it doesn't move us any closer to an explanation of why there are any elephants at all.

Suppose there is a finite chain of elephants, with the first elephant being the descendant of two non-elephants. (That's basically what we have in the actual world, modulo worries about vagueness of elephanthood.) Then only one link in that chain does any work in explaining why there are any elephants: the link from the two non-elephants to the elephants.

Now suppose we have an infinite chain of elephants. No link in the chain does any work in explaining why there are any elephants. Sure, the links explain all sorts of things: They explain why Dumbo exists, why Dumbo's parents do, and so on. But what they don't explain is why there are any elephants at all.

We can say what sort of thing could explain that. For instance, a non-elephant that initiated the whole infinite chain of elephants (either from beyond time, or from another time sequence, or from a time before all the infinitely many finite past times). But why there are any elephants isn't explained by any or all the elephants in the chain.

This shows an interesting difference between the disjunction and conjunction questions. Let P be a conjunction of all the elephant-existence claims (Dumbo exists, Dumbo's mother exists, etc.) Then each of the links in an infinite chain does contribute to a partial explanation of P. Dumbo's parents' mating contributes to explaining one conjunct of the giant conjunction. But (a) together these links do not suffice to explain all of P, because (b) they all leave untouched the disjunction question--why there are any elephants at all.

Angra Mainyu said...

Okay, disjunctions are a different matter. At this point, I think that there some issues that would have to be addressed, such as:

a. The dependence of explanations and agents. For instance, if Bob is a hunter-gatherer with no modern technology who asks Alice to explain how is it that there are images in a computer screen, an account in terms of Quantum Physics wouldn't seem to provide an explanation.
b. The dependence of explanations and contexts, even for the same agent. The question: "Why does Dumbo exists?" may well be answered in some contexts by an account in terms of his parents. However, if the person asking the question had last seen his parents a decade ago (Dumbo is younger), and his parents were about to be killed by human hunters, the proposed explanation may not be acceptable (better examples can be found, if needed),


That aside, in the case of a Dumbo, the account in terms of his parents does not fully explain why he exists. Nor does it explains it to someone who does not know that elephants are the kind of things that reproduce sexually, so that a male and a female elephant can produce an elephant.

On the other hand, let's say that someone proposes an explanation for each of the elephants, not only in terms of other elephants, but in terms of other stuff, that makes the explanation sufficient for the agent asking the question. For instance, all elephants at time t, for each t, are explained by some events at some time t' < t, some of which involved elephants. If the person asking for an explanation accepts all of that, it's not clear to me that the question has not been answered.

If the person asking for an explanation accepts the explanation for each of the elephants, I do not see why she would reject the explanation of the conjunction in terms of the explanations of the conjuncts.
So, even if some disjunction remained unexplained, I would say that all of P has been explained, as long as the explanation E(A) is sufficient to explain each elephant A in that context.

Angra Mainyu said...

Regarding disjunctions, I'm not sure what disjunction you have in mind, but I get the question is: 'Why are there elephants at all?'.
So, let's say that the person asking the question is not satisfied by an answer in terms of the explanations E(A) as to why each elephant A exists, and yet she accepts the explanation E(A) as a good explanation of the existence of that particular elephant A; let's say the explanation involves not only elephants but particles and stuff.

I'm not entirely sure what else she would be asking for, in that case, though in practice, given that we know about elephants in particular, infinite regress of elephants shouldn't and probably wouldn't be accepted, and I think that that might play a role in some intuitions about your scenario.

As for a non-elephant beyond time, I do not see why it would explain the chain. I have to admit I do not understand how something "beyond time" initiates a chain, so that explanation would not work for me, or for a number of other people.

In the end, it seems to me that given the kind of question you're considering, someone might ask something like: "Why are there concrete objects at all?"

If no explanations in terms of concrete objects succeed, then the question has no answer.
If, on the other hand, an explanation in terms of concrete objects explains why there are concrete objects at all, a question is what kind of an explanation.
As I see it, infinite regress would be as good as it gets. If that Intuitively, I would say that that explains it as much as it could be explained, and if that fails, then that's it – i.e., there is no explanation.
Granted, theists often offer an explanation in terms of a necessary concrete object. But I do not find a reply like 'A concrete object exists necessarily' to be an explanation as to why concrete objects exist at all.
In terms of possible worlds, that strikes me as the following reply: Let's say that Bob asks why there are dolphins on planet Earth. Alice replies that there are dolphins on all Goldilocks planets, and Earth is one such planet. That surely would not seem to work. But the claim that a being exists in all possible worlds does not seem to explain why it exists in each world as far as I can tell (yes, okay, someone might say that her nature is such that if she exists, she exists necessarily; I'm afraid I can't make sense of existing in one and/or more worlds being part of anything's nature).
In other words, as I see it, in that case, not only is the fact that such being exists [assuming she does] an inexplicable puzzle, but also the fact that at all other possible worlds, she exists as well.
If possible worlds talk is not accepted, still saying that some concrete object exists necessarily wouldn't explain why she exists as far as I can tell (i.e., intuitively I do not find that to explain it), but would raise the further puzzle of why it exists necessarily.

Still, if a necessary concrete object (assuming she exists) ought to be accepted (assuming for the sake of the argument) as an explanation as to why concrete objects exist, then what it seems to me that would be in need of an explanation is why concrete objects necessarily exist at all, and that is intuitively a lot more puzzling to me than the mere actual existence of concrete objects.

Yes, granted, many (plausibly most) theists make different intuitive assessments. So, it seems to me that either intuitions will converge given sufficient reflection and rationality, or whether there might be an explanation depends on the intuitions of specific agents.

ashmen Dasgupta said...

I've been looking for this article for awhile and it isn't on your Selected Papers and Essays. How can one get access to this paper or others like it as my library and school do not carry this journal as this looks like a very good paper. Anyways thanks!

Alexander R Pruss said...

The usual thing to do when one can't find a copy of a paper is to email the author a request. They are generally very happy to oblige.

Hassan Abdillah said...

Dr. Pruss,

I'm sure you've considered this before, but what do you think about the contra-Hume argument that explaining contingent things in terms of other contingent things and the entire chain of explanation being infinite fails to explain anything at all (since each explanation attempt leaves the explanation itself unexplained), and hence the only hope for a complete explanation is positing a Necessary Being's action?

The only person I've seen using this argument is Paul Herrick, here: http://infidels.org/library/modern/paul_herrick/parsons.html#regress

It makes sense to me at an intuitive level, except the concern that PSR or the CP may not be strong enough to entail such an "absolute" conception of explanation. What do you think?