Devil's arithmetic, or a contribution to angelology
The famous problem of "the number of devils that can go on a pin's point" has indeed been seriously considered in the past, although this way of stating it is only due to Renaissance mockery. The man who was very serious about it was Thomas Aquinas who dwelt upon it in his Summa Theologiae, part II (On Angels), chapter LII, section 3: "Can several angels be simultaneously in one and the same place?" You may feel surprised with the change: was it finally about devils or angels? There is no change at all! Every devil is also an angle, as follows from the writings of the same Thomas of Aquino, called "doctor angelicus" because of his deep knowledge of angels, included in Summa Theologiae in chapters LXII section 1 and LXIII sections 8 and 9. There is, however, an important modification which may have escaped your attention: in one case the statement mentions a pin's point, in the other it involves the notion of a place. The difference is essential: a pin's point is meant to represent a point (in the geometric sense), while of the notion of place Aquinas says in Summa Theologiae LII.2:
"On this, however, some have erred. For being unable to go beyond their imagination, they assumed the angel's indivisibility in analogy to the indivisibility of a point and this induced them to think that an angel can only appear in a place which is a point. But then it is clear that they are in the wrong; for a point is something indivisible endowed with a position, whereas angel's indivisible existence is beyond any kind of quantity or situation. Therefore it is not necessary that an angel have one determined place indivisible in position; on the contrary, it can be divisible or indivisible, bigger or smaller, depending on where the angel chooses by free will to apply its power to a bigger or smaller body."
In view of the above opinion we shall consider a place to be any subset of the three-dimensional Euclidean space.
To solve the problem of the number of angels being in one place we must first define what is meant by "being" in a place. Doctor angelicus says: "...an angel is in a place by applying its power to this place..." The same idea is expressed even more strongly by John of Damascus (II de Fid. Orth., cap.3): "... the angel is where it acts." In fact, Thomas of Aquino quotes this citation in Summa Theologiae LII.2. Thus we can say the following:
(1) An angel is at a place X when it is the cause of the events occurring in that place.
The main question is answered by Aquinas as follows: "... two angels cannot simultaneously exist in one place" (ibidem, sec.3). The reasoning is the following (ibidem):
(*) "And the reason for it is the fact that it is impossible for two complete causes to be direct causes of one and the same thing. This is clear with respect to any kind of causes..."
To explain the details of this argument and in particular to explain why "this is clear...", let's turn to Aristotle (Physics II.3), where a classification of causes is included. The Philosopher lists the following causes of events:
1] Material causes. A material cause is the matter of which the effect is made. Aristotle illustrates the idea by the example of a stone statue: its material cause is the stone. Clearly,
(2) An angel cannot be the material cause of bodily phenomena, for it is bodiless itself.
In his announcement of chapter L of Summa Theologiae Thomas writes: "Next we should consider...the purely spiritual being which in the Holy Scriptures is given the name of angel..."
2] Formal causes. This is the form adopted by the effect. In the case of the statue it is its shape. It seems natural to assume that
(3) An angel cannot be the formal cause of bodily phenomena.
3] Efficient causes. This is the being or the phenomenon which brings forth the effect. For the statue it is the sculptor or the sculptor together with the patron who orders the work, or more precisely, the sculptor's decision to produce the statue.
4] Final causes. This is why the event occurs. The final cause of the statue is to embellish the park.
Let's go back to Thomas's argument. We have to prove that only one angle can be at a particular place, i.e. we must prove that if the angels A and B are at a place X, then A=B and A and B are one and the same. So, let's assume A and B are at a place X. This simply means that the angels A and B are causes for the phenomena occurring at X. By the assumptions (2) and (3) three cases are possible:
a] Both angels are efficient causes. This seems to be implied by the use of the word "complete" in (*). Aquinas seems to assume the following:
(4) If A is an efficient cause of phenomena at X and B is an efficient cause of phenomena at X, then A and B are one and the same.
b] Both angels are final causes. In this case the argument relies upon some laws of purposeful actions:
(5) Every action has a purpose.
(6) Every action has only one purpose.
Hence if two angels were to be final causes of events at a place X, then two actions would have to occur at X, which in turn implies that both angels would be efficient causes at X. This, however, has already been excluded.
c] One of the angels is an efficient cause and the other is a final cause.
We believe that this case had been overlooked by Thomas. There is not sufficient ground in Summa Theologiae to make the corresponding assumption. Maybe at this point Doctor Angelicus committed an error in his argument?
Marcin Mostowski, Leslaw W. Szczerba