Classical mereology and causal regresses

Assume classical mereology with arbitrary fusions.

Further assume two plausible theses:

1. If each of the ys is caused by at least one of the xs and there is no overlap between any of the xs and ys, then the fusion of the ys is caused by a part of the fusion of the xs.

2. It is impossible to have non-overlapping objects A and B such that A is caused by a part of B and B is caused by a part of A.

It follows that:

3. It is impossible to have an infinite causal regress of non-overlapping items.

For suppose that A₀ is caused by A₋₁ which is caused by A₋₂ and so on. Let E be a fusion of the even-numbered items and O a fusion of the odd-numbered ones. Then by (1), a part of E causes O and a part of O causes E, contrary to (2).

This is rather like explanatory circularity arguments I have used in the past against regresses, but it uses causation and mereology instead.

Probabilities of regresses of chickens

Suppose we have a backwards-infinite sequence of asexually reproducing chickens, ..., c₋₃, c₋₂, c₋₁, c₀ with c_n having a chance p_n of producing a new chicken c_n + 1 (chicken c₀ may or may not have succeeded; the earlier ones have succeeded). Suppose that the p_n are all strictly between 0 and 1, and that the infinite product p₋₁p₋₂p₋₃... equals some number p strictly between 0 and 1.

Intuitively, we should be surprised that chicken c₀ exists if p is low and not surprised if p is high. If we have observed c₀ and are considering theories as to what the chances p_n are, other things being equal, we should prefer the theories on which the product p is high to ones on which it’s low.

But what exactly does p measure? It seems to be some kind of a chance of us getting c₀. But it doesn’t measure the unconditional probability of getting an infinite sequence of chickens leading up to c₀. For that is very tiny indeed, since it is extremely unlikely that the world would contain chickens at all. It seems to be a kind of conditional probability. Let q_n be the proposition that chicken c_n exists. Then P(q₀∣q_n) = p₀p₋₁p₋₂...p_n, and so p is the limit of the conditional probabilities P(q₀∣q_n). It is plausible thus to think of p as a conditional probability of q₀ on q_−∞, which is the infinite disjunction of all the q_n.

But q_−∞ is a rather odd proposition. It is grounded in q_n for every finite n, assuming that a disjunction, even an infinite one, is grounded in its true disjuncts. Thus every one of the q_n is explanatorily prior to q_−∞. But this means that P(q₀∣q_−∞) is actually a conditional probability of q₀ on something that isn’t explanatorily prior to q₀—indeed, that is explanatorily posterior to q₀. This challenges the interpretation of p as a chance of getting chicken c₀.

I am not quite sure what conclusion to draw from the above argument. Maybe it offers some support for causal finitism, by suggesting that things are weird when you have a backwards infinite causal sequence?

More on God causing infinite regresses

In my previous two posts I focused on the difficulty of God creating an infinite causal regress of indeterministic causes as part of an argument from theism to causal finitism. In this post, I want to drop the indeterministic assumption.

Suppose God creates a backwards infinite causal regress of (say) chickens, where each chicken is caused by parent chickens, the parent chickens by grandparent chickens, and so on. Now, I take it that the classical theist tradition is right that no creaturely causation can function without divine cooperation. Thus, every case where a chicken is caused by parent chickens is a case of divine cooperation.

Could God’s creative role here be limited to divine cooperation? This is absurd. For then God would be creating chickens by cooperating with chickens!

So what else is there? One doubtless correct thing to say is this: God also sustains each chicken between its first moment of life and its time of death. But this sustenance doesn’t seem to solve the problem, because the sustenance is not productive of the chickens—it is what keeps each chicken in existence after it has come on the scene. So while there is sustenance, it isn’t enough. God cannot create chickens by cooperating with chickens and by sustaining them.

Thus God needs to have some special creative role in the production of at least some of the chickens, fulfilling a task over and beyond cooperation and sustenance. Furthermore, this special task must be done by God in the case of an infinite number of the chickens, since otherwise there would be a time before which that task was not fulfilled—and yet God created infinitely the chickens before that time, too, since we’re assuming an infinite regress of chickens.

What happens in these cases? One might say is that in these special cases, God doesn’t cooperate with the parent chickens. But since no creaturely causation happens without divine cooperation, in these cases the parent chickens don’t produce their offspring, which contradicts our assumption of the chickens forming a causal regress. So that won’t do.

So in these cases, we seem to have two things happening: divine cooperation with chicken reproduction and divine creation of the chicken. Since divine cooperation with chicken reproduction is sufficient to produce the offspring, and divine creation of the chicken is also sufficient, it follows that in these cases we have causal overdetermination.

Now, we have some problems. First, does this overdetermination happen in all cases of chicken reproduction or only in some? It doesn’t need to happen in all of them, since it is overdetermination after all. But if it happens only in some, then it is puzzling to ask how God chooses which cases he overdetermines and which he does not.

Second, when there is overdetermination, the overdetermination is not needed for the effect. So it seems that if God’s additional role is that of overdetermining the outcome, that role is an unnecessary role, and the chickens could be produced by mere divine cooperation, which we saw is absurd. This isn’t perhaps the strongest of arguments. One might say that while in each particular case the overdetermining divine creative action is not needed, it is needed that it occur in some (indeed, infinitely many) cases.

Third, just as it is obviously absurd if God creates chickens merely by cooperating with chickens, it seems problematic, and perhaps absurd, that God creates chickens merely by cooperating with chickens and overdetermining that cooperation.

Famously, Aquinas thinks that God could have created an infinite regress of fathers and sons, and hence presumably of chickens as well. At this point, I can think of only one plausible way of getting Aquinas out of the above arguments, and it’s not a very attractive way. Instead of saying that God cooperates with the production of offspring, we can say that occasionalism holds in every case of substantial causation, that all causation of one substance’s existence by another is a case of direct divine non-cooperative causation, with the creaturely causation perhaps only limited to the transmission of accidents. Like all occasionalism, an occasionalism about substance causation is unappealing philosophically and theologically.

Partial and complete explanations

1. Any explanation for an event E that does not go all the way back to something self-explanatory is merely partial.

2. A partial explanation is one that is a part of a complete explanation.

3. So, if any event E has an explanation, it has an explanation going all the way back to something self-explanatory. (1,2)

4. Some event has an explanation.

5. An explanation going back to something self-explanatory involves the activity of a necessary being.

6. So, there is an active necessary being. (4,5)

I am not sure I buy (1). But it sounds kind of right to me now. Additionally, (3) kind of sounds correct on its own. If A causes B and B causes C but there is no explanation of A, then it seems that B and C are really unexplained. Aristotle notes that there was a presocratic philosopher who explained why the earth doesn’t fall down by saying that it floats on water, and he notes that the philosopher failed to ask the same question about the water. I think one lesson of Aristotle’s critique is that if it is unexplained why the water doesn’t fall down it is unexplained why the earth falls down.

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.

Gunk, etc.

If we think parts are explanatorily prior to wholes, then gunky objects—objects which have parts but no smallest parts—involve a vicious explanatory regress. But if one takes the Aristotelian view that wholes are prior to parts, then the regress involved in gunky objects doesn’t look vicious at all: the whole is prior to some parts, these parts are prior to others, and so on ad infinitum. It’s just like a forward causal regress: today’s state causes tomorrow, tomorrow’s causes the next day’s, and so on ad infinitum.

On the other hand, on the view that parts are explanatorily prior to wholes, upward compositional regresses are unproblematic: the head is a part of the cow, the cow is a part of the earth, the earth is a part of the solar system, the solar system is a part of the Orion arm, the Orion arm is a part of the Milky Way, the Milky Way is a part of the Local Group, and this could go on forever. The Aristotelian, on the other hand, has to halt upward regresses at substances, say, cows.

This suggests that nobody should accept an ontologically serious version of the Leibniz story on which composition goes infinitely far both downward and upward, and that it is fortunate that Leibniz doesn’t accept an ontologically serious version of that story, because only the monads and their inner states are to be taken ontologically seriously. But that's not quite right. For there is a third view, namely that parthood does not involve either direction of dependence: neither do parts depend on wholes nor do wholes depend on parts. I haven't met this view in practice, though.

Tropes of tropes

Suppose that x is F if and only if x has a trope of Fness as a part of it.

Here is a cute little problem. Suppose Jim is hurting and has a trope of pain, call it Pin. But Pin is an improper part of Pin. Thus, Pin has a trope of pain—namely itself—as a part of it, and hence Pin is hurting. Thus, wherever someone is hurting, there is something else hurting, too, namely their pain.

The standard move against “two many thinkers” moves is to say that one of them is thinking derivatively. But if we do that, then it looks like the fact that Jim is hurting is more likely to be derivative than the fact that Pin is hurting. For Jim hurts in virtue of having Pin as a part of it, while Pin hurts in virtue of having itself as a part of it, which seems a non-derivative way of hurting. But it seems wrong to say that Jim is hurting merely derivatively, so the real subject of the pain is Pin.

An easy solution is to say that x is F if and only if x has a trope of Fness as a proper part of it.

But this leads to an ugly regress. A trope is a trope, so it must have a trope of tropeness as a proper part of it. The trope of tropeness is also a trope, so it must then have another trope of tropeness as a proper part and so on. (This isn’t a problem if you allow improper parthood, as then you can arrest the regress: the trope of tropeness has itself as an improper part, and that’s it.)

One can, of course, solve the problem by saying that the trope theory only applies to substances: a substance x is F if and only if x has a trope of Fness as a proper part of it, while on the other hand, tropes can have attributes without these attributes being connected with the tropes having tropes. But that seems ad hoc.

As a believer in Aristotelian accidents and forms, which are both basically tropes, I need to face the problem, too. I have two ways out. First, maybe all tropes are causal powers. Then we can say that if “is F” predicates a power, then x is F if and only if x has a trope of Fness as a proper part. But for attribution of non-powers, we have a different story.

Second, maybe the relation between objects and their tropes is not parthood, but some other primitive relation. Some things stand in that relation to themselves (maybe, a trope of tropeness stands in that relation to itself) and others do not (Pin is not so related to itself). This multiplies primitive relations, but only if the relation of parthood is a primitive relation in the system.

A regress of qualitative difference

According to heavyweight Platonism, qualitative differences arise from differences between the universals being instantiated. There is a qualitative difference between my seeing yellow and your smelling a rose. This difference has to come from the difference between the universals seeing yellow (Y) and smelling a rose (R). But one doesn’t get a qualitative difference from being related in the same way to numerically but not qualitatively different things (compare: being taller than Alice is not qualitatively different from being taller than Bea if Alice and Bea are qualitatively the same—and in particular, of the same height). Thus, if the qualitative difference between my seeing yellow and your smelling a rose comes from being related by instantiation to different things, namely Y and R, then this presupposes that the two things are themselves qualitatively different. But this qualitative difference between Y and R depends on Y and R exemplifying different—and indeed qualitatively different—properties. And so on, in a regress!

Counting down from infinity

In one version of the Kalaam argument, Bill Craig argues against forming an infinite past by successive addition by asking something like this: Why would someone who had been counting down from infinity have been finished today rather than, say, yesterday? This argument puzzles me. After all, there is a perfectly good reason why she finished today: because today she reached zero and yesterday she was still on the number one. And yesterday she was on one because the day before she was on two. And so on.

Of course, one can object that such a regress generates no explanation. But then the Kalaam argument needs a Principle of Sufficient Reason that says that there must be explanations of such regressive facts and an account of explanation according to which the explanations cannot be found in the regresses themselves. And with these two assumptions in place, one doesn’t need the Kalaam argument to rule out an infinite past: one can just run a “Leibnizian style” cosmological argument directly.

Infinite grounding regresses

Suppose, as seems possible, that every day for eternity you will toss a coin and get heads.

Then that you will get heads on every future day seems to be grounded in:

1. You will get heads on day 1, and you will get heads on every day starting with day 2.

And the second conjunct of (1) seems to be grounded in:

2. You will get heads on day 2, and you will get heads on every day starting with day 3.

And the second conjunct of (2) seems to be grounded in:

3. You will get heads on day 3, and you will get heads on every day starting with day 4.

And so on.

So, it seems, infinite propositional grounding regresses are possible.

I suspect that infinite existential grounding regresses are not possible, though.

Positing non-epistemic vagueness doesn't solve a puzzle

Suppose we want to explain why one tortoise doesn't fall down, and we explain this by saying that it's standing on two tortoises. And then we explain why the two lower tortoises doesn't fall down, we suppose that each stands on two tortoises. And so on. That's terrible: we're constantly explaining one puzzling thing by two that are just as puzzling.

Now suppose we try to explain the puzzle of the transition from bald to non-bald in a Sorites sequences of heads of hair (no hair, one hair, two hairs, etc.). We do this by saying that there are going to be vague cases of baldness. But this is just as the case of tortoises. For while previously we had one puzzling transition, from bald to non-bald, now we have two puzzling transitions, from definitely bald to vaguely bald and from vaguely bald to definitely bald. So, we repeat with higher levels of vagueness. The transition from definitely bald to vaguely bald yields a transition from definitely bald to vaguely vaguely bald and a transition from vaguely vaguely bald to definitely vaguely bald, and similarly for the transition from vaguely bald to definitely bald. At each stage, each transition is replaced with two. We're constantly explaining one puzzling thing by two that are just as puzzling.

That said, it is possible with care to stand a tortoise on two tortoises, and we could have evidence that a particular tortoise is doing that. In that case, the two tortoises aren't posited to solve a puzzle, but simply because we have evidence that they are there. A similar thing could be the case with baldness. We might just have direct evidence that there is vagueness in the sequence. But as we go a level deeper, I suspect the evidence peters out. After all, in ordinary discourse we don't talk of vague vagueness and the like. So perhaps we might have a view on which there is one level of vagueness--and then epistemicism, i.e., there is a sharp transition from definitely non-bald to vaguely bald, and another from vaguely bald to definitely bald. But the more levels we posit, the more we offend against parsimony.

A virtuous evidential regress

Could this ever be the case: p₂ is evidence for p₁, p₃ is evidence for p₂, p₄ is evidence for p₃, and so on ad infinitum?

I don't think we can rule this out on epistemological grounds alone. For suppose that there are infinitely many unicorns in the universe, none of which you've observed, but there are also infinitely many experts. Expert number n happens to inform you that there are at least n unicorns in the universe. Now, let p_n be the proposition that there are at least n unicorns in the universe. Then obviously p₁ is evidence for p₂, p₃ is evidence for p₂ and so on. But there is nothing vicious about this regress. For you have independent evidence for each p_n. This is a case where although there is an infinite evidential regress, all the ultimate evidence is outside of the regress—for ultimately all the evidence about the unicorns comes from the experts.

But note that despite the fact that the ultimate evidence is all outside the regress, the evidential relations within the regress are important. For while you have some evidence for p₁ directly from the first expert, you also have some additional evidence for p₁ deriving from p₂, and hence from the second expert.

Infinite dependence regresses and set theory

Given the Axiom of Dependent Choice, the Axiom of Regularity in set theory is equivalent to the statement that there are no backwards infinite membership regresses, i.e., no cases where we have a backwards infinite sequence of sets ...,A₋₃,A₋₂,A₋₁,A_-0, where each set is a member of the next. Why think this is true? Well, intuitively, a set depends on its members. That suggests that the reason to believe the Axiom of Regularity is that there cannot be an infinite dependency regress. And that in turn has all sorts of other consequences (including that there is a first cause).

Against per se ordered infinite sequences of causes

Say that a sequence of events is per se causally ordered provided that each event not only causes the next but also causes everything in the next that is involved in causing the one after that (if there is one after that).

1. Any possible chunk of contingent reality is such that it is possible for something internally just like it to have a cause.
2. It is not possible to have a cause for an infinite per se causal regress of contingent causes.
3. Anything internally just like an infinite per se causal regress of contingent causes is an infinite per se causal regress of contingent causes.
4. So an infinite per se causal regress of contingent causes is impossible.

In the argument, (1) is a weak causal principle. The reason for the "something internally just like" phrase is that without the phrase the premise would immediately imply that every possible chunk of contingent reality has a cause given essentiality of origins. Premise (3) requires a non-Humean account of causation. I am going to ignore metaphysical questions about chunks of reality: perhaps we can reformulate in terms of pluralities, perhaps in terms of sets.

A crucial controversial premise is (2). Here's an intuitive line of thought that inclines me to (2). Suppose we have a backwards-infinite per se causal sequence of chickens and eggs, each chicken fully deterministically causing an egg with all of its relevant causal power, and each egg deterministically causing a chicken with all of its relevant causal power. And imagine this regress has a cause, say, G. How can G cause that whole sequence? Well, it couldn't do it by causing one particular chicken or one particular egg, for that wouldn't account for the chickens and eggs that came before that. The only picture I get of how something could cause the whole sequence would be if it caused each one of the eggs and chickens, or each one prior to some point in time, or more generally some backwards-infinite subsequence. The argument will be the same in each of the three cases, so I will just focus on the simplest.

So the cause G caused each of the eggs and chickens, and thereby caused the sequence. But of course each egg is caused by a chicken, too. So a given egg has two causes: it is caused by a chicken and by G. But the chicken caused the egg fully, with everything the egg needed to do its job of causing the next thing. So what did G contribute? Nothing really crucial to the sequence, since everything crucial to it was contributed by the chicken. Rather, it looks like it's going to be a case of overdetermination. The egg is caused by G and it's caused by the chicken. But G's causing of the regress as a whole isn't overdetermined (or so we may surely assume, modulo some technicalities), since in the absence of G the whole sequence wouldn't be caused. And if you take away a non-overdetermining cause, the effect disappears. So if you took away G, the whole regress should disappear. But why should taking away G matter, given that all of the causal influences by which G allegedly causes the regress are individually overdetermined?

If this is all right, then the critical attention will shift to (1).

System-relativity of proofs

There is a generally familiar way in which the question whether a mathematical statement has a proof is relative to a deductive system: for a proof is a proof in some system L, i.e., the proof starts with the axioms of L and proceeds by the rules of L. Something can be provable in one system—say, Euclidean geometry—but not provable in another—say, Riemannian geometry.

But there is a less familiar way in which the provability of a statement is relative. The question whether a sentence p is provable in a system L is itself a mathematical question. Proofs are themselves mathematical objects—they are directly the objects in a mathematical theory of strings of symbols and indirectly they are the objects of arithmetic when we encode them using something like Goedel numbering. The question whether there exists a proof of p in L is itself a mathematical question, and thus it makes sense to ask this question in different mathematical systems, including L itself.

If we want to make explicit both sorts of relativity, we can say things like:

1. p has (does not have) a proof in a system L according to M.

Here, M might itself be a deductive system, in which case the claim is that the sentence "p has (does not have) a proof in L" can itself be proved in M (or else we can talk of the Goedel number translation of this), or M might be a model in which case the claim is that "p has a proof in L" is true in that model.

This is not just pedantry. Assume Peano Arithmetic (PA) is consistent. Goedel's second incompleteness theorem then tells us that the consistency of PA cannot be proved in PA. Skipping over the distinction between a sentence and its Goedel number, let "Con(PA)" say that PA is consistent. Then what we learn from the second incompleteness theorem is that:

2. Con(PA) has no proof in PA.

Now, statement (2), while true, is itself not provable in PA.[note 1] Hence there are non-standard models of PA according to which (2) is false. But there are also models of PA according to which (2) is true, since (2) is in fact true. Thus, there are models of PA according to which Con(PA) has no proof and there are models of PA according to which Con(PA) has a proof.

This has an important consequence for philosophy of mathematics. Suppose we want to de-metaphysicalize mathematics, move us away from questions about which axioms are and are not actually true. Then we are apt to say something like this: mathematics is not about discovering which mathematical claims are true, but about discovering which mathematical claims can be proved in which systems. However, what we learn from the second incompleteness theorem is that the notion of provability carries the same kind of exposure to mathematical metaphysics, to questions about the truth of axioms, as naively looking for mathematical truths did.

And if one tries to de-metaphysicalize provability by saying that what we are after in the end is not the question whether p is provable in L, but whether p is provable in L according to M, then that simply leads to a regress. For the question whether p is provable in L according to M is in turn a mathematical question, and then it makes sense to ask according which system we are asking it. The only way to arrest the regress seems to be to suppose that at some level that we simply are talking of how things really are, rather than how they are in or according to a system.

Maybe, though, one could say the following to limit one's metaphysical exposure: Mathematics is about discovering proofs rather than about discovering what has a proof. However, this is a false dichotomy, since by discovering a proof of p, one discovers that p has a proof.

Not a finetuning argument

In The Impiety... (1624), as part of the 6th argument for the existence of God, Mersenne writes:

The proportion found between all the bodies of the world also shows that there is a God who has made all the universe in weight, in number and in measure: for the earth has no other ratio with the sun than 1:140, with the moon than 40:1, ... (pp. 98-99)

(I don't know off hand what the ratios are exactly meant to be; if they are ratios of volume, the moon is within 25% of the truth but the sun is off several orders of magnitude; if they are ratios of diameter, the sun is within an order of magnitude of the truth but the moon is an order of magnitude off.)

Mersenne's argument is full of such numerical (claimed) facts (the sun goes around the earth in 365.241 days, the moon traverses the Zodiac in 27 days, etc., etc.) and claims that God is needed to explain these facts. Now, I'm right now teaching on the fine-tuning argument, so I am sensitized to seeing such numbers in an argument for the existence of God. But it's striking that nowhere can I see Mersenne saying why these numbers are at all better than others, especially since surely some tuning facts seem very close at hand--surely, for instance, if the sun were much bigger or much smaller than it is, it would be too hot or too cold for life.

Mersenne explicitly insists that the numbers aren't explained by the essential natures of the objects, just before the above quote:

For the sun wouldn't be any the less the sun if it were closer or further from the earth, just as the stars could still be stars if they absented themselves from us by more than 14,000 earth radii.

Mersenne's argument seems to be a pure application of the idea that all contingent facts need explanation, and the arbitrariness of the numbers in the numerical statements seems to be cited precisely in order to show the contingency of the numerical statements. The argument suggests a strikingly strong commitment to a Principle of Sufficient Reason for contingent facts: all he needs to argue for a cosmic cause is to argue that there are contingent cosmic facts. Mersenne is confident that God has "many reasons" (as he says in the case of one of the numerical claims) for making the numbers be what they are, but these are reasons "which we aren't going to know except in Paradise" (101-102).

Mersenne's argument isn't a design argument--it doesn't advert to value-laden features that a God would have good reason to actualize. I think it's a kind of cosmological argument, but an eccentric one. Rather than arguing from generic features like motion or causation as Aquinas did, it focuses on very particular features.

The focus on these very particular features seems to have two benefits. The first is that it makes any appeal to necessity as the explanation implausible. Maybe it's necessary that there is motion, but it is incredible that it be necessary that the ratio of the diameter of the earth to that of the moon have to be 3.665:1 (to use modern numbers). So we get contingency very easily. The second feature is one I didn't notice right away. The astronomical features cited by Mersenne are ones that would reasonably be thought to be permanent features. They are thus prime candidates to be dismissed by it is so, as it has always been so. Mersenne's focus on the seeming arbitrariness of these features makes it very clear that would be no explanation. Thus Mersenne's cosmological argument works whether or not the past is finite. It is not disturbed by an infinite regress but does not need one either.

Of course, we no longer think that these particular features are permanent in the same way--the earth and sun changed in size in the formation of the solar system. But impermanent features are no better explained by an infinite regress than permanent ones--the permanence of the features in Mersenne's argument is only heuristic (and I don't see him explicitly drawing the reader's attention to the permanence). Plus we could run the argument on the basis of the apparently permanent but seemingly arbitrary elements in the laws of nature, such as precise values of constants.

The downside of Mersenne's argument, however, is that unless it is explained why the features are desirable, it is difficult to show that the cause of these features of the universe must be intelligent.

Infinite regress explanations

Consider Thomson's toggle lamp—each time the button is pressed, the lamp toggles between on and off—but suppose it existed from eternity and every January 1 the switch has been pressed once, and only then. Why is the lamp on now? Consider the regress explanation: It's on in 2014 because it was off in 2013 and toggled on January 1, 2014. And it was off in 2013 because it was on in 2012 and toggled on January 1, 2013. And so on.

Hume will say that this is a complete explanation. But surely not. Surely the whole story does not explain why the lamp is on in even numbered years and off in odd numbered years.

Notice an interesting thing. The following are perfectly fine explanations:

1. The lamp is on in 2014 because it was off in 2013 and toggled at the beginning of 2014.
2. The lamp is on in 2014 because it was on in 2012 and toggled at the beginnings of 2013 and 2014.
3. The lamp is on in 2014 because it was off in 2011 and toggled at the beginnings of 2012, 2013 and 2014.

And as we go down this list of explanations, our explanations get more and more ultimate. However, we can't take this to infinity. Each of the explanations in the list has wo conjuncts: a fact about the state of the lamp in year n, and then facts about the lamp being toggled in successive years. The facts about the lamp being toggled in successive years can be taken to infinity, but aren't enough to explain it. The following clearly isn't enough to give us an ultimate explanation of why the lamp was on in 2014:

4. The lamp was toggled at the beginnings of ..., 2010, 2011, 2012, 2013 and 2014.

Can we take the first conjunct in explanations (1)-(3) to infinity? Well, we certainly can't in general say that the lamp was on, or that it was off, in year −∞, since even if such a year existed, dubious as that is, the lamp need not have existed then—it need only be supposed to exist in all finite-numbered years. So what can we say? Well, we could let the lamp state in year n be L(n)—0 being off and 1 being on—and then say:

5. The limit of L(2n) is 1 as n→−∞ and the limit of L(2n+1) is 0 as n→−∞ (both limits over the integers only).

So if we think about how to complete our regressive explanation, it seems that it will need to be something like this:

6. The lamp is on in 2014 because of (4) and (5).

Very good. But even if (4) were to be ulitimately explained (maybe there is some mechanism where each toggling is caused by the preceding, which according to Hume would give an ultimate explanation of (4)), it is clear that (5) calls out for an explanation as well, and so the regressive explanation just isn't ultimate explanation.

So infinite regresses aren't enough for ultimate explanations, pace Hume.

Another argument against an infinite past?

I wonder if this very neat argument can't be used to provide another Grim Reaper style argument against an infinite past? The argument nicely fits with the intuition that Grim Reaper induces in me, namely that no event can have an infinite number of events in its causal history.

Actual infinity

I am occasionally asked what I think about actual infinities.  Mainly because of a Grim Reaper argument, but also because of the bob argument here, I do think there is a problem about certain kinds of actual infinities.  But not all of them.  The Grim Reaper argument, as well as the bob argument, only tells against one kind of actual infinity: an actual infinity of causal influences on a single event.  So while one could conclude that actual infinities are impossible, a more conservative move is to conclude that it is not possible for an infinite number of causal influences to be in the past history of any event.

This limited anti-infinitism means that I do not need to worry about arguments in favor of actual infinities such as the following:

• The actual existence of an infinite future.
• The actual existence of mathematical infinities.
• The intuitively very plausible possibility of a simultaneous infinity of objects.  At least, it's plausible to me.

In fact, my worry is not so much about an actual infinite as such, but about infinitely many causal influences coming together.  

Initially, when I came to the Grim Reaper argument against an infinite past, I found the anti-infinitist conclusion very counterintuitive.  But now it has become clear to me that it's not: for it's not that counterintuitive to think that absurdities can result if an infinite number of causal influences can work together.  My pro-infinitist intuitions were based on non-causal mathematical considerations.  But I can, if I wish, retain those intuitions.  

In fact, I can even retain the intuition that there could have been an infinite past, as long as this does not imply a backwards-infinite causal chain.  Consider, for instance, a world that consists of a multiverse of universes.  The first universe is one year old.  The second universe is two years old.  The third is three years old.  And so on.  The world as a whole has an infinite past.  But as long as there isn't the wrong kind of causal dependence between the universes, such an arrangement need not imply the kind of infinity in causal influence that my arguments lead me to think is problematic.

This strengthens the Kalaam argument by showing that the premises can be weakened: the Kalaam argument only needs the kind of causal anti-infinitism that I now cautiously accept.

Collimating a collimator

Collimating a Newtonian telescope basically means aligning the optical axis of the primary mirror with the optical axis of the eyepiece.  An easy way to do this is to use a collimator, e.g., a laser collimator.  The collimator is basically a tube that contains a laser that you put in place of the eyepiece.  You adjust the angle of the secondary mirror in the telescope so the beam hits the center of the primary mirror, and then you adjust the angle of the primary mirror so that the beam comes back on itself.  But it is crucial for this procedure with a laser collimator that the collimator be itself collimated, i.e., that the laser's axis be aligned with the tubing that the laser is in.

Now, if we were writing a philosophy paper, at this point it would be very tempting to say: "And a vicious infinite regress ensues."  But that would too quick.  For a laser-collimator collimator is very simple: a block of wood with two pairs of nails, where each pair makes an approximate vee shape.  You then lay the laser collimator on the two vees, aim it at a fairly distant wall, and spin it.  Then you adjust the adjustments screws on the laser collimator until the beam doesn't move as you spin the collimator on its axis, at which point the laser is collimated to its housing.  Moreover, because of how the geometry works, the vees don't need to be very exactly parallel--all the work is done by spinning.  So, it seems, the regress is arrested: the laser-collimator collimator does not itself need collimation.

Potential lesson: Perhaps sometimes we philosophers are too quick after one or two steps in a regress to say that the regress is vicious and infinite.  For sometimes after two steps, the regress may be stopped with a bit of cleverness.

Well, actually, that's not quite right.  For the double-vee collimator depends on the laser collimator's housing being a cylinder.  And one might argue that manufacturing an exact cylinder requires a procedure like collimation.  For suppose that we manufacture the cylinder by taking a block of aluminum, spinning it in a lathe and applying a lathe tool.  But to get an exact cylinder, the lathe tool needs to remain, at the end, at an equal distance to the lathe's rotational axis.  So that's another alignment procedure that's needed.  I don't know how that's done, being foggy on the subject of lathes, but I bet it involves aligning some sort of a guide parallel to the lathe's rotational axis or by moving the workpiece parallel to the rotational axis.  So another collimation step will then be needed when manufacturing the lathe.

And so the regress does continue.  But still only finitely.  At some point, parallelism can be achieved, within desired tolerance, by comparing distances, e.g., with calipers.  There is still a collimation issue for the calipers, but while previous collimations involved the spatial dimensions, the collimation of calipers uses spatial and temporal dimensions: in other words, the calipers must keep their geometrical properties over time.  For instance, if one sets the calipers to one distance, and then compares another, the caliper spacing had better not change over the amount of time it takes the calipers to move from one place to another.  So calipers allow one to transfer uniformity over time into uniformity over space.

But how do we ensure uniformity over time?  By using a rigid material, like hardened steel.  And how do we ensure the rigidity of a material?  This line of questioning pretty quickly leads to something that we don't ensure: laws of nature, uniform over space and time, that make the existence of fairly rigid materials possible.  And if we then ask about the source of these laws and their uniformity, the only plausible answer is God.  So, we may add to God's list of attributes: ultimate collimator.

There are, of course, other ways of manufacturing cylinders than by using a lathe.  One might cast a cylinder in a cylindrical mould--but that just adds an extra step in the regress, since the mould has to be manufactured.  Or one might extrude a cylinder by pushing or pulling the material for it through a circular die.  In the latter case, one still has to make a circular die, perhaps with a spinning cutter at right angles to a flat piece, and one has to ensure that the material is moved at right angles to the die.  So one has changed the problem of parallelism into the very similar problem of aligning at right angles.  And I suspect we eventually get back to something like rigid materials anyway.

So the lesson that sometimes regresses stop after one or two steps is not aptly illustrated with the case of the collimator.  That regress is still, perhaps, finite--but it goes further back, and eventually to God.