I avoided saying much about the key part of the argument, which was the insistence that causal sequences cannot proceed to infinity. To refresh our memories, the argument goes like this:
1) Since everything in motion is moved by some other thing, andThe entire argument rests on Step 2), as the first step is relatively uncontroversial. Things don't pop in and out of existence. Things don't move for no reason whatsoever. Everyone should be on board so far. But the Step 2) is really troublesome. What exactly is Aquinas claiming?
2) since movers and things moved cannot proceed to infinity,
3) therefore, we must posit the existence of an Unmoved Mover, which we call God.
There are two immediate possibilities. The first says, An infinite regression is not a plausible explanation for anything, in the same way that to say the world rests on the back of a elephant, which stands on the back of a turtle, and thereafter its turtles all the way down, is not an good explanation for what keeps the world fixed in place. The second says, An actual infinite something (mass, volume, causal series, etc.), whatever it might be and however that might be measured, cannot exist, in a manner similar to our universe, which is for all intents and purposes practically infinite, from our limited human vantage, but shows physical evidence of having finite limits, in particular, a temporal beginning that set everything motion. (For good measure, let's give the beginning a name and call it the Big Bang, after the well known comedy show The Big Bang Theory, which made geekdom and nerdom cool.)
Deciding which is the better interpretive option is not easy. When Aquinas says, 'movers and things moved cannot proceed to infinity', it appears to be an instance of the second possibility. The plain meaning of the phrase points to an actual infinity. However, as soon as you ask the question, why 'movers and things moved cannot proceed to infinity', you must go looking for reasons. The sort of reasons that lend themselves to answering the question are more likely instances of the first possibility. An infinite causal series cannot exist because an infinite regression is not a plausible explanation for anything.
Now, a possible argument against the existence of an actual infinite causal series of movers and things moved offered by the 6th century, Christian Neo-Platonist critic of Aristotle, John Philoponnus, goes as follows. If the causal series is infinite, then there has been an infinite number of movers and things moved up to the present moment and will be a second infinite number of movers and things moved after the present moment. That means means there will exist two infinities, which is contradictory, and so also false, because there can only ever be one infinity
It's unlikely that Aquinas, however, was so numerically inclined. Numbers had been the obsession of Platonists (1+2+3=6), not Aristotelian's like Aquinas, who were more interested in thinking through the natural forms of causal relations (A follows B follows C, which means A follows C). It's only with Isaac Newton these two modes of explanation get brought together into a single system of The Mathematical Principles of Natural Philosophy (1687), rendering the systematic relations between quantities like velocity (m/s), acceleration (m/s2), and force (kg*m/s2) intelligible.
What then might Aquinas be on about when he says 'movers and things moved cannot proceed to infinity'? Aquinas assumes that the human mind is attuned to the natural order of things, such that if it is mentally true, it is also true of extra-mental things, and vice versa. Via their intellect, human beings participate, in some way, in the intelligence which calls the world into existence.
Now it is true that the human mind cannot process an infinite causal series. If you don't believe me, go ahead and try it for yourself. Chances are you mentally charted out an A followed by a B, B followed by C, C followed by D, and D followed by...the infinite series itself. Note carefully that your attention shifted from individual steps in an infinite causal series to the causal series itself because it wasn't possible for you to chart out every single step in the series. Because if, in fact, you did try, you are still trying, and will still be trying, and never succeeding to comprehend an infinite series in thought, until the day you die. Might as well stop while your ahead.
So if it's true, as Aquinas holds, the human intellect participates in the intelligence that calls the world into existence, and if it's true that what applies for one also applies for the other, it will also be true that 'movers and things moved cannot proceed to infinity'.