Per se and per accidens multiplication of causes

Can there be an infinite sequence of efficient causes? Famously, Aquinas says both “No” and “Yes”, and makes a distinction between a per se ordering (“No”) and an accidental ordering (“Yes”). But it is difficult to reconstruct how the distinction goes, and whether there is good reason to maintain given modern physics.

Here is the central passage from Summa Theologiae I.46.2 reply 7, in Freddoso’s translation:

It is impossible to proceed to infinity per se among efficient causes, i.e., it is impossible for causes that are required per se for a given effect to be multiplied to infinity—as, for instance, if a rock were being moved with a stick, and the stick were being moved by a hand, and so on ad infinitum.

By contrast, it is not impossible to proceed to infinity per accidens among agent causes, i.e., it is not impossible if all the causes that are multiplied to infinity belong to a single order (ordinem) of causes and if their multiplication is incidental (per accidens)—as, for instance, if a craftsman were to use many hammers incidentally, because one after another kept breaking. In such a case, it is incidental to any given hammer that it acts after the action of a given one of the other hammers. In the same way, it is incidental to this man, insofar as he generates, that he himself was generated by another. For he generates insofar as he is a man and not insofar as he is the son of some other man, since all the men who generate belong to the same order (gradum) of efficient causality, viz., the order of a particular generating cause. In this sense, it is not impossible for man to be generated by man ad infinitum.

However, it would indeed be impossible for the generation of this man to depend upon that man, and upon an elemental body [a corpore elementari], and upon the sun, and so on ad infinitum.

What’s going on here? Re-reading the text (and double-checking against the Latin) I notice that per se and per accidens are introduced not as modifying the causal relations, but the infinite multiplication of causes. No indication is given initially that the causation functions differently in the two cases. Further, it is striking that both of the examples of per accidens multiplication of causes involve causes of the same type: hammers and humans (Freddoso’s “man” translates homo throughout the text).

To a first approximation, it seems then that what is forbidden is a regress of infinitely many types of causes, whereas a regress of infinitely many tokens is permitted. But that is too simple. After all, if an infinite causal sequence of humans generating humans were possible, it would surely also be possible for each of these humans to be qualitatively different from the others—say, in exact shade of eye color—and hence for there to be infinitely many types among them. In other words, not just any type will do.

Let’s focus in on two other ingredients in the text, the observation that the humans all “belong to the same order of efficient causality”, and the sun–elementary body–human example. Both of these rang a bell to me, because I had recently been writing on the Principle of Proportionate Causality. At Summa Theologiae I.4.2, St Thomas makes a different distinction that distinguishes between the human–human and the sun–body–human cases:

whatever perfection exists in an effect must be found in the effective cause: either in the same formality, if it is a univocal agent—as when man reproduces man; or in a more eminent degree [eminentiori modo], if it is an equivocal agent—thus in the sun is the likeness of whatever is generated by the sun’s power.

Here is a suggestion. In distinguishing per se and per accidens infinite multiplication of causes, Aquinas is indeed distinguishing counting types and tokens. But the types he is counting are what one might call “causal types” or “perfections”. The idea is that we have the same causal type when we have univocal agency, “as when man reproduces man”, and different causal type when we have equivocal agency, as when the sun generates something, since on Aquinas’ astronomical theory the sun is sui generis and hence when the sun generates, the sun is quite different from what it generates. In other words, I am tentatively suggesting that we identify the gradus of efficient causality of I.46.2 with the modus of perfection of I.4.2.

The picture of efficient causation that arises from I.4.2 is that in a finite or infinite causal regress we have two types of moves between effect and cause: a lateral move to a cause with the same perfection as the effect and an ascending vertical move to a cause that has the perfection more eminently.

The lateral moves only accidentally multiply the explanations, because the lateral moves do not really explain the perfection. If I got my humanity from another human, there is a sense in which this is not really an explanation of where my humanity comes from. The human I got my humanity from was just passing that humanity on. I need to move upwards, attributing my humanity to a higher cause. On this reading, Aquinas is claiming that there can only be finitely many upwards moves in a causal regress. Why? Maybe because infinite passing-on of more to less eminent perfections is just as unexplanatory as finite passing on of the same perfection. We need an ultimate origin of the perfections, a highest cause.

I like this approach, but it fits better with the sun–elemenatary body–human example than the hand–stick–rock example. It seems, after all, that in the hand–stick–rock example we have the same relevant perfection in all three items—locomotion, which is passed from hand to stick and then from stick to rock. This would thus seem like a per accidens multiplication rather than a per se one. If so, then it is tempting to say that Aquinas’ hand–stick–rock example is inapt. But perhaps we can say this. Hand-motion is probably meant to be a voluntary human activity. Plausibly, this is different in causal type from stick-motion: going from stick to hand is indeed an explanatory ascent. But it’s harder to see the progression from rock to stick as an explanatory ascent. After all, a rock can move a stick just as much as a stick can move a rock. But perhaps we can still think we have an ascent from rock-moving to stick-moved-by-hand, since a stick-moved-by-hand maybe has more of the perfection of the voluntary hand motion to it? That sounds iffy, but it’s the best I can do.

I wish Aquinas discussed a case of stick–stick–stick, where each stick moves the next? Would he make this be a per se multiplication of causes like the hand–stick–rock case? If so, that’s a count against my reading. Or would he say that it’s an accidental multiplication? If so, then my tentative reading might be right.

It’s also possible that Aquinas’ examples of hand–stick–rock and sun–elementary body–human are in fact more unlike than he noticed, and that it is the latter that is a better example of per se multiplication of causes.

A Thomistic argument for the Principle of Proportional Causality

The Principle of Proportionate Causality (PPC) defended by Aquinas and other scholastics says that a perfection P can only be caused by something that has P either formally or eminently. To have P formally is to have P. Roughly, to have P eminently is to have a perfection greater than P.

(Some add: “has P virtually” to the list of options. But to have P virtually is just to have the power to produce P, and as our student Colin Causey has noted, this trivializes PPC.)

There are obvious apparent counterexamples to PPC:

• Two parents who are bad at mathematics can have a mathematical genius as a child.

• Ugly monkeys typing at random can produce a beautiful poem.

• A robot putting together parts at random can make a stronger and smarter robot.

It’s tempting to throw PPC out. But there are also cases where one feels a pull towards PPC:

• How can things that represent come from non-representing stuff?

• How can the conscious come from the non-conscious?

• How can something with dignity come from something without any?

• How can the active come from the inactive?

• How can an “ought” come from a mere “is”, i.e., something with normativity from something without any?

Many contemporary philosophers think there is no impossibility even in these cases, but I think most will agree that there is something puzzling about these kinds of causation—that we have some sort of an intuition towards PPC in these cases, of a sort we do not have in the cases of the “obvious apparent counterexamples”. What is the difference between the cases?

Well, in the counterexamples, the differences between the cause and the effect are, arguably, a matter of degree. The two parents have a much lower degree of mathematical ability. The monkeys have a certain beauty to them—being productive of beauty is a kind of beauty—albeit perhaps a lesser one than their lucky output. The robot’s output is just a more sophisticated bunch of moving parts than the robot itself.

But in the examples where one feels pulled to PPC, the differences appear to be differences in kind. Indeed, I think we can all agree that the most plausible way to resist the implied claim in the “How can…?” questions that the thing is impossible is to show how to reduce the seemingly more perfect thing to something of the same sort as the alleged cause.

But “differences in kind” doesn’t seem quite sharp enough. After all, pretty much everyone (even, I assume, young earth creationists) will agree that dogs can come from wolves.

I’ve been puzzled by how one might understand and argue for PPC for a long time, without much progress. This morning I had an inspiration from Nicholas Rescher’s article on Aquinas’ “Principle of Epistemic Disparity”, that lesser minds cannot comprehend the ways of greater ones.

Suppose we order the types of good by a comprehensibility relation ≤ where G ≤ H means that it is possible to understand G by understanding H. Then ≤ is a partial preorder, i.e., a reflexive and transitive relation. It generates a strict partial preorder < where G < H provided that G ≤ H but not H ≤ G.

Next, say that good types G₁ and G₂ are cases of the same perfection provided that G₁ ≤ G₂ and G₂ ≤ G₁, i.e., that each can be understood by the other. Basically, we are taking perfections to be equivalence classes of types of good, under the relation ∼ such that G₁ ∼ G₂ if and only if G₁ ≤ G₂ and G₂ ≤ G₁. The relation ≤ extends in a natural way to the perfections: P ≤ Q if and only if whenever G is a case of P and H is a case of Q then G ≤ H. Note that ≤ is a partial order on the perfections. In particular, it is antisymmetric: if we have P ≤ Q and Q ≤ P, then we have P ≠ Q. Write P < Q provided that P ≤ Q and P ≠ Q.

Now on to a Thomistic argument for the PPC.

Being, truth and goodness are transcendentals. The cognitively more impressive perfection Q is thus also axiologically more impressive. Thus:

Axiological Thesis: If P < Q for perfections P and Q, then Q is a better kind of perfection than P.

The following is plausible on the kind of Aristotelian intrinsic notion of causation that Thomas works with:

Causal Thesis: By understanding the cause one understands the effect.

Thomistic ideas about transcendentals also yield:

Understandability Lemma: To understand a thing one only needs to understand the goods instantiated by the thing.

Finally, let’s add this technical assumption:

Conjunction Lemma: The conjunction of co-instantiable goods is a good.

And now on to the PPC. Suppose x causes y to have a good G and y has a type of good G that is a case of a perfection P. By the Causal Thesis, we understand G by understanding x. By the Conjunction Lemma, let H be the conjunction of all the good of x. By the Understandability Lemma, we understand x by understanding H. Thus, G ≤ H. Let Q be the perfection that H is a case of. Then P ≤ Q and x has Q. Then either P = Q or P < Q. In the former case, the cause has P formally. In the latter case, by the Axiological Thesis, the cause has P eminently.

Of course, the Axiological and Causal Theses, together with the Understandability Lemma, all depend on large and controversial parts of Aquinas’ system. But I think we are making some progress.

I am also toying with an interesting concept. Say that a perfection Q is irreducible provided that it cannot be understood by understanding any conjunction of perfections P such that P < Q. It’s not obvious that there are irreducible perfections, but I think it is plausible that there are. If so, one might have a weaker PPC restricted to irreducible perfections. I have yet to think through the pluses and minuses here.

Proportionate causality

Let’s assume for the sake of argument:

Aquinas’ Principle of Proportionate Causality: Anything that causes something to have a perfection F must either have F or some more perfect perfection G.

And let’s think about what follows.

The Compatibility Thesis: If F is a perfection, then F is compatible with every perfection.

Argument: If F is incompatible with a perfection G, then having F rules out having perfection G. And that’s limitive rather than perfect. Perhaps the case where G = F needs to be argued separately. But we can do that. If F is incompatible with F, then F rules out all other perfections as well, and as long as there is more than one perfection (as is plausible) that violates the first part of the argument.

The Entailment Thesis: If F and G are perfections, and G is more perfect than F, then G entails F.

Argument: If F and G are perfections, and it is both possible to have F without having G and to have F while having G, it is better to have both F and G than to have just G. But if it is better to have both F and G than to have just G, then F contributes something good that G does not, and hence we cannot say that G is more perfect than F—rather, in one respect F is more perfect and in another G is more perfect.

From the Entailment Thesis and Aquinas’ Principle of Proportionate Causality, we get:

The Strong Principle of Proportionate Causality: Anything that causes something to have a perfection F must have F.

Interesting.