A famous parody of Scholasticism is, of course, about the question, "How many angels can dance on the head of a pin?"
We don't really know that much about angels from Revelation. Ott lists only 10 dogmas regarding angels, and though we know in general of their nature as pure spirits, their test, their fall, their role in man's fall, and their role in man's salvation, in a general sense, I think we have very little sense of the specific "lives" of the angels. And rightly so. Their "dispensation" is really different than ours (though totally intertwined, of course, in the divine plan).
Given that each angel must be (as Aquinas says) a totally different and unique species with a radically individual nature, though existing along a sort of "continuum," because they have no matter to differentiate them otherwise...it would be rather hard to speak of the "specifics" of angelic nature given that there is no one "species" or "nature" to speak of, but an incredible absolute diversity. Every angelic consciousness experiences things in a totally unique "language," apparently. If we can assume that, though human brains differ, all human consciousness has the same basic capacity and experiences the same basic qualia...angels exist along a continuum of knowledge, each its own slightly different species (in the scholastic sense).
Now, I'm not sure where the "dancing on the head of a pin" came from, but the "official" answer to the basic question of how many angels can be in a location is, in fact, given by Aquinas and is quite reasonable, actually.
Not having bodies, angels do not occupy space in the same sense as matter, obviously, with dimensive quality. However, inasmuch as angels are thought able to apply a certain power to matter, they may be said to occupy however much space that to which they are applying their power occupies, as St. Thomas explains:
An angel's power and nature are finite, whereas the Divine power and essence, which is the universal cause of all things, is infinite: consequently God through His power touches all things, and is not merely present in some places, but is everywhere. Now since the angel's power is finite, it does not extend to all things, but to one determined thing. For whatever is compared with one power must be compared therewith as one determined thing. Consequently since all being is compared as one thing to God's universal power, so is one particular being compared as one with the angelic power. Hence, since the angel is in a place by the application of his power to the place, it follows that he is not everywhere, nor in several places, but in only one place. Some, however, have been deceived in this matter. For some who were unable to go beyond the reach of their imaginations supposed the indivisibility of the angel to be like that of a point; consequently they thought that an angel could be only in a place which is a point. But they were manifestly deceived, because a point is something indivisible, yet having its situation; whereas the angel is indivisible, and beyond the genus of quantity and situation. Consequently there is no occasion for determining in his regard one indivisible place as to situation: any place which is either divisible or indivisible, great or small suffices, according as to his own free-will he applies his power to a great or to a small body. So the entire body to which he is applied by his power, corresponds as one place to him.This, of course, suggests an answer to the "dancing on the head of a pin" question:
There are not two angels in the same place. The reason of this is because it is impossible for two complete causes to be the causes immediately of one and the same thing. This is evident in every class of causes: for there is one proximate form of one thing, and there is one proximate mover, although there may be several remote movers. Nor can it be objected that several individuals may row a boat, since no one of them is a perfect mover, because no one man's strength is sufficient for moving the boat; while all together are as one mover, in so far as their united strengths all combine in producing the one movement. Hence, since the angel is said to be in one place by the fact that his power touches the place immediately by way of a perfect container, as was said, there can be but one angel in one place.However, the head of a pin is not necessary just one "place." As many angels can be said to dance on the head of the pin as the pin-head can be conceived of as separate objects (say, individual atoms or molecules) and thus separate places. Just one angel might apply his power to the whole pin-head as one object (or even to the whole room which contains it), and then no other angels could "dance" on it. Or he might pick "the left half of the pinhead" leaving the right-half open to one angel, two angels (if they divide it further into specific quarters), or as many other angels as there are potential divisions into discrete objects of angelic power.
Now, this was all just for fun. I think it's interesting and, though certainly not the subject of Revelation, certainly seems like an incredibly sensible and precise speculation by Aquinas for showing that Revelation is not in conflict with Reason, at least. That there is at least one internally consistent way to interpret these things and answer these questions. Some people may hate this sort of speculative theology or think that it contributes nothing to our salvation or is just trivial or mental masturbation. I think it's fun though.
As long as we don't get dogmatic about it, I see nothing wrong with these sorts of thought experiments and scholastic puzzles. I think they, at the very least, incline us to appreciate the beauty of God's creation and to meditate in some sense on whatever it is we are reasoning about. As an aside, I'll just point out, that Aquinas discusses a fascinating theory from Augustine about the "days" of Genesis 1 being really "days of angelic knowledge." The "morning" knowledge being the angels perceiving the Forms of material things in the Eternal Word, and the "evening" knowledge being the progression to their aeviternal perception of these things' actual existence, perceived (at least by the highest angels, whose understandings are most universal) in a logical series of the six categories laid out in Genesis for the six days. It's certainly something I think Biblical literalist-fundamentalists (and those who accuse the Catholic Church of having been so until recently) should read!
This is not what my conversation was about however. The part of my conversation that got me thinking about the head of a pin example was about how many angels there are. The person I was talking to was questioning the power of prayer on (oddly) both the grounds of the determinism of the universe and human free will. Saying that people use prayer to "excuse" themselves from acting or to make themselves feel like they've done something when they should be out there actively helping. The objection boiled down to that prayer's effect can't be scientifically verified, that if we did an empirical experiment, the group that prayed wouldn't get what they were praying for any more often than the group which wasn't.
I'd actually be the first to concede this is probably true. It's not because I don't believe in the power of prayer. There was a time when, exactly due to that same sort of empiricist mindset, I may have doubted that it had a real material effect very often, and tended to aver to its positive spiritual effects; but I've very much been shown its real practical power lately too. However, I do not believe its effects could ever be demonstrated by anything like empirical study or statistics derived from experiments because I believe it involves simply a very different sort of causation.
The person I was talking to was upset that someone suggested to him that her prayers got him a job. He was "offended" even, given that "she did nothing except light a candle, while I was out there applying for jobs, working hard to get them." However, I pointed out, that's only one tiny aspect. The other is, of course, enormously one of circumstantial opportunity. Yes, you had to be prepared and take initiative of course, but even that's no guarantee. An incredible path of coincidences has to line up for us to do anything, to complete any task, from the cars we pass on the road, to the signs we see diverting our train of thought. It is in these "coincidences" that I think Catholics would tend to see the absolutely sovereign hand of Providence and, thus, also the site of the power of prayer.
It's like, I was reading about Evans-Pritchard's writing about the Azande people of Africa once, and their "magical thinking" that modern empirically-minded Westerners often find frustrating:
The classic example is of the collapsing roof, described in E. E. Evans-Pritchard's Witchcraft, Magic, and Oracles Among the Azande, in which the Azande claimed that a roof fell on a particular person because of a magical spell cast (unwittingly) by another person.
The Azande knew perfectly well a scientific explanation for the collapsing room (that termites had eaten through the supporting posts), but pointed out that this scientific explanation could not explain why the roof happened to collapse at precisely the same moment that the particular man was resting beneath it. The magic explains why two independent chains of causation intersect. Thus, from the point of view of the practitioners, magic explains what scientists would call "coincidences" or "contingency". From the point of view of outside observers, magic is a way of making coincidences meaningful in social terms. Carl Jung coined the word synchronicity for experiences of this type.
And if it's according to His Will, then He might. "Well, mightn't have happened even without the prayer?" Sure. But just like invoking the Saints, God seems to like to work through secondary causes, and if our prayer asks for something, then even God in His Providence is ultimately the immediate cause, He seems happy to share the credit if we "contributed" in anyway by our prayers. Just because a strong man can pull a block by himself, that does not mean that if a child pulls on the rope too he is contributing nothing or is not then part of the cause of the movement.
Will prayer always work? No, not in a straightforward sense of getting what we want. Sometimes what we're praying for isn't in accord with His will. And sometimes, yes, it would have just happened anyway. But we have to assume that, often enough at least, the very act of prayer will make it more likely, vis a vis the maximizing salvation and merit (the end goal of Providence, really), for God to arrange it. Of course, the very act of prayer might occasionally make it more likely not to happen (say, if God is trying to teach us to accept His Will or something) when it would have happened if we hadn't prayed. But Revelation and experience seem to tell us that if we pray it is much more likely to increase the chances of something happening than to decrease it.
In this sense, I said, prayer (and magic) is really more about trying to get the probabilities to lean in your favor a bit more, trying to shift the probabilities, through appeal to the cause of coincidence. I even mentioned the experiments (I'm not sure they were ever conclusive) where humans merely concentrating on the output of random number generators seemed to suggest that the random distribution could be skewed ever so slightly, statistically speaking, towards whichever outcome the person was focusing on.
The next objection from this particular agnostic was that God can't control the universe, which we know is run by deterministic laws. All this "arranging" of events, he would say, makes no sense. There is no mechanism whereby it could be true if things have been "playing out" mechanistically since the beginning of the Universe. However (after a side conversation on predestination) this is where I pointed out that his science is actually bad here. Quantum Mechanics has proven (as in the double-slit experiment) that the material universe is actually probabilistic, not deterministic.
There are different intepretations of this data, of course, from the Copenhagen interpretation that "consciousness causes collapse" of the wave function (though how, given that we don't seem to explicitly choose our reality, is a major question), to the Many Worlds hypothesis which says every time a particle makes one of those undeterministic "choices"...a different universe is created with each possibility (which leaves major questions about why our consciousness is in this universe instead of the other, or even about what "to exist" means if the "parallel" universe cannot effect ours or our consciousness in any way). Both "interpretations" are really two sides of the same coin, I think, suggesting that this is where science ends and philosophy begins, specifically a philosophy requiring subjectivity, subjective consciousness.
Basically, my suggestion here about Providence was that the "mechanism" by which God could be sovereign over history would be that He is the one deciding which of the probabilistic possibilities (when particles make these otherwise inscrutable "choices") is actualized. It might also be then speculated that the connection between our spirits and our bodies, by which free will might actually affect the body rather than consciousness being a mere epiphenomenon...might be that we are given the power to (in an indirect way) control the quantum "choices" within our own brains. This would seem to allow for interactionism rather than parallelism, occasionalism, or epiphenomenalism.
This is where we get back to angels. I mentioned that the Medievals attributed the movement of the heavenly bodies, for example, not to simply impersonal physical laws being enforced (why the constants of physics are universal is always another question that science can't account for except by appealing to philosophy). Rather, they believed that angels or choirs of angels moved the heavenly spheres. That God mediated the causation through these spirits who did what He wanted. I mentioned then, as an aside, that in the "providence by way of quantum decision-making" theory, it doesn't have to be God directly controlling the particles. That it would not be beyond the realm of possibility to believe that each particle or group of particles, in that regard, were controlled by an angelic consciousness.
However, the number of particles in the universe (and just the visible universe at that?) has been estimated at between 10 ^ 72 and 10 ^ 87. That's 10 followed by 72-87 zeroes! Which is perhaps not unsurprising given an estimate of hundreds of billions of galaxies, each with hundreds of billions of stars. It is truly a spectacularly huge place (the question of intelligent life on other planets is a question for another time...) And are we really to believe that there are 10^87 angels?
Aquinas might actually be inclined to think so. The Summa article about "How many angels are there?" says:
Hence it must be said that the angels, even inasmuch as they are immaterial substances, exist in exceeding great number, far beyond all material multitude. This is what Dionysius says (Coel. Hier. xiv): "There are many blessed armies of the heavenly intelligences, surpassing the weak and limited reckoning of our material numbers." The reason whereof is this, because, since it is the perfection of the universe that God chiefly intends in the creation of things, the more perfect some things are, in so much greater an excess are they created by God. Now, as in bodies such excess is observed in regard to their magnitude, so in things incorporeal is it observed in regard to their multitude. We see, in fact, that incorruptible bodies, exceed corruptible bodies almost incomparably in magnitude; for the entire sphere of things active and passive is something very small in comparison with the heavenly bodies. Hence it is reasonable to conclude that the immaterial substances as it were incomparably exceed material substances as to multitude.However, I am a bit uncomfortable with this. Perhaps I am too much of a humanist, but I have always tended to link the total number of angels with the total number of human beings. In fact, I've tended to speculate that the total number of humans will be "complete" when it equals the total number of the angels. I'm not alone in linking the two, as this excerpt from Catholic Encyclopedia on the total number of the saved shows:
Naturally, human curiosity is eager for definite information about the absolute as well as the relative number of the elect. How high should the absolute number be estimated? But it would be idle and useless to undertake calculations and to guess at so and so many millions or billions of predestined. St. Thomas (I, Q. xxiii, a. 7) mentions the opinion of some theologians that as many men will be saved as there are fallen angels, while others held that the number of predestined will equal the number of the faithful angels. Lastly, there were optimists who, combining these two opinions into a third, made the total of men saved equal to the unnumbered myriads of created spirits. But even granted that the principle of our calculation is correct, no mathematician would be able to figure out the absolute number on a basis so vague, since the number of angels and demons is an unknown quantity to us. Hence, "the best answer", rightly remarks St. Thomas, "is to say: God alone knows the number of his elect". By relative number is meant the numerical relation between the predestined and the reprobate. Will the majority of the human race be saved or will they be damned? Will one-half be damned the other half saved? In this question the opinion of the rigorists is opposed to the milder view of the optimists [...] The truth is that neither the one nor the other can be proved from Scripture or Tradition (cf. Heinrich-Gutberlet, "Dogmat. Theologie", Mainz, 1897, VIII, 363 sq.). But supplementing these two sources by arguments drawn from reason we may safely defend as probable the opinion that the majority of Christians, especially of Catholics, will be saved [...] then Gener is probably right when he assumes the salvation of half of the human race, lest "it should be said to the shame and offense of the Divine majesty and clemency that the [future] Kingdom of Satan is larger than the Kingdom of Christ" (cf. W. Schneider, "Das andere Leben", 9th ed., Paderborn, 1908, 476 sq.).Of course, if exactly half the human race is imagined as being saved, and exactly half the angels imagined as falling (the "one third of the stars" from Revelations aside)...then the humans who are saved could be imagined as both "matching" the number of saved angels, as well as "filling up the spots" of the fallen angels. Then every human being and angel or demon could be imagined in a one-to-one pairing with each other, either in heaven (their guardian angel on earth) or in hell (their familiar demon). Such a theory requires every angel (and demon) being assigned to two humans over the course of history, of course, one elect and one reprobate, in order for the numbers to work out so that every angel and every demon winds up with someone.
This is where my conversation wandered. I concluded that such "numerical" speculation about the angels and demons relative to human saved and damned...was very "medieval" in its "symmetry," very elegant in that way, but also that life is perhaps more messy than this, not actually "symmetrical," as much as the medievalist in me has a tendency to wish it were. And, of course, such a "half will be saved" idea excludes the hope for universal salvation which I nurture (then again, if even the demons somehow could be saved, things could work out just as symmetrically!)
Still, I think that even if we hope that all will be saved, imagining it as exactly half is spiritually healthy. It prevents us from either despairing or presuming. If we imagine, at least, that exactly half will be saved (whether that's true or not is really none of our business and irrelevant), but if we imagine it that way...it makes us aware of how our salvation is really our choice. Then we can neither "fall back on" the idea that "Oh, most people are saved, so I'll be fine," nor despair with the idea that "Oh, most people are damned, so the tables are naturally set against me." If we think of it as half, then we cannot plead the probabilities. Then we have an absolutely free choice to make.
As St. Augustine himself is alleged to have once said (though the attribution may only be as recent as Samuel Beckett and Waiting for Godot), "Do not despair; one of the thieves was saved. Do not presume; one of the thieves was damned." Whether the latter actually was damned or not (I hope not), the message for our souls is clear: the proportions in terms of other people don't matter at all; act as if the "chances" neither favor you nor disfavor you. And I think that's very important advice.