All of Aquinas's Five Ways depend, in one way or another, on ruling out the possibility of infinite causal regresses. In the version of the First Way (the argument from motion) in the Summa Contra Gentiles (I.13), Aquinas follows Aristotle in offering two separate arguments against the causal regress. In the first three ways in the Summa Theologiae, he offers just one of the two Aristotelian arguments: an argument that depends on what I call the No-Intermediate-Real-Cause thesis. This is a thesis that states that if x is a cause of y, and y is a cause of z, then y is not really a cause in the strict sense but only secundum quid (only in a manner of speaking). An intermediate link in a causal chain is not in any sense the source of the reality of the ultimate effect--it is merely a conduit through which the first cause acts. Therefore, an infinite regress is impossible, because (as Aristotle and Aquinas note) every link in the regress would be only an intermediate cause. Hence, such a regress cannot contain any real causation.
 
This is a plausible argument, despite the fact that it is often dismissed as based on a fallacy of equivocation. The standard objection (going back at least to Cajetan, I believe) is that Aquinas equivocates on the phrase "removing the first cause." If we have a finite chain and we hypothetically remove the first cause from the series, it is obvious that none of the intermediate causes can act. Aquinas asserts that if we consider any infinite regress, we have a situation from which we have (in a sense) "removed the first cause". But, as critics point out, in this case there never was a first cause to be "removed", and so the cases are incomparable. However, Aquinas real point is simply to claim that intermediate causes are never causes in their own right but are always wholly parasitic on the first cause. Given that assumption, infinite causal regresses are indeed impossible.
 
The other strategy for dealing with infinite regresses was invented by Avicenna, followed by Scotus, Leibniz, and many subsequent thinkers (including me in 1997). This is the Aggregation Strategy. The Aggregation Strategy concedes, for the sake of argument, that infinite regresses are possible. However, the Strategy insists, if a regress consists entirely of contingent (or finite) things, then we can aggregate the whole series into a single entity and insist on a cause for it. Start now with a single finite thing, and consider all of the finite causes of that thing (whether immediate or remote). Either this series terminates in an uncaused thing, or else it constitutes an infinite regress. In the latter case, we can demand a cause for the whole series. This cause must be infinite, since any finite cause of the series would be a cause of the original entity and so would already be included in the series itself. A member of the series cannot cause the whole series. An infinite thing cannot be caused. And so we reach an uncaused first cause.
 
Aquinas was aware of this strategy, through his close reading of Avicenna. Why didn't he adopt it? I think he was worried that not all infinite series can be aggregated into a single entity. If an infinite series consists of entities of the same species (or a finite number of species), then it has the characteristic that Aquinas labels being accidentally infinite. An accidentally infinite series can be aggregated and must be caused as a whole. This is why Aquinas can concede that accidentally infinite series might exist without losing the force of his first cause argument. However, if an infinite series consists of entities of an infinite number of species, with the species climbing progressively higher and higher in the Great Chain of Being, the Aggregation Strategy could not be convincingly applied. Aquinas would call such a series essentially infinite, and he must (if the Second or Third Way is to work) deny the metaphysical possibility of such a series. This is why he appeals to Aristotle's argument and what I call the No-Intermediate-Real-Cause thesis, which should now be applied only to series that rise "vertically" through the ontological order of species.

​Aquinas's second way, the way of efficient causation, had perhaps the most influence on subsequent natural theology. It has counterparts in the work of Scotus, Leibniz, and Samuel Clarke (to name a few), and it is the argument that Kant labels "cosmological" in the First Critique. I am going to assume the argument that Aquinas develops in chapters 3 and 4 of De Ente et Essentia is an elaboration of the second way. The argument's form is quite simple:
 
1. Some things actually exist (as known primarily by sense perception).
2. Every finite thing (i.e., thing for which there is a real distinction between essence and existence) that actually exists has an actually existing cause of its existence.
3. Causes are prior to their effects in the order of actual existence.
4. Every chain of causes has an essential structure.
5. The essential structure of every chain of causes is finite (has terminal, uncaused nodes).
6. Therefore, there exists at least one actually existing infinite being, and every finite thing is caused by one or more such beings.
7. There can be no more than one infinite being.
8. Therefore, there exists exactly one actually infinite being, which is the ultimate cause of every finite being.
 
If something has actual existence, and this actual existence is something distinct from its essence, then there is some part of the thing's essence which is disjoint from its existence. The finite thing's existence depends on a joining or combining of its existence with the remainder of its essence, and this joining or combining must have some explanation. It cannot be explained by the remainder of the essence, since, it it were, the thing would be a cause of itself (contrary to premise 3). It cannot be caused by the thing's actual existence, since a thing's existence cannot be prior to any part of its essence, since every 'act' or 'event' of existence depends for its very identity on the essence that it actualizes. Hence, the only possible explanation for this joining or combining must appeal some distinct entity (its efficient cause).
 
In contrast, if a thing's existence is identical to its essence, no explanation of its existence is needed or even possible. The essence of a thing constitutes its possible existence, and if a thing's existence is identical to its essence, then that thing's actuality is identical to its possibility. Hence, we cannot sensibly ask how or why its possibility has been actualized. For such a thing to be really possible is for it to be actual, and vice versa. If such a being exists at all, it must exist with absolute necessity. For this reason, the causal principle (premise 2) must be limited to finite things.
 
Premise 7: There can be only one infinite thing (so defined). Suppose that there were two. We can ask whether it is possible for one to exist without the other. If so, then one or the other exists only contingently, which we've have shown to be impossible. So, it must be necessary for both to exist and to relate to each other as distinct entities. Is this relation of distinctness contained in the essence of one or both? It couldn't be just one, since, if it were, the other would depend for its existence on the other. So, the relation must be contained in both essences. But this would introduce some complexity into both essences, which means that we could no longer identify either essence with a simple act of existence. In addition, there would have to be some explanation of the symmetry. Each would have to depend essentially on the existence of another, again contrary to their simplicity and uncausability.
 
This leaves us with premises 4 and 5, which jointly rule out the possibility of an essentially infinite regress.  I will take up this crucial question next time.