How Many Angels Can Dance on the Head of a Pin?
Though it’s doubtful that medieval angelologists ever debated about how many angels can dance on the head of a pin, but, as Peter Kreeft says, “It’s a good question.”
Let’s look at this problem more closely.
What is an angel?
An angel is a “separated substance” other than a human soul after death but before the resurrection of the body, that is, an angel is a pure spirit. He is not a soul because he is not meant to inform a body. An angel is a substantial form with no matter of any kind. His essence and existence are not identical, that is, he is not God but rather a creature.
What, then, does it mean for an angel to be in a place?
Since an angel is a spirit, he is not in a place as bodies are in a place. A body is necessarily accidentally in a place by virtue of what it is to be a body. A spirit, however, has no matter and cannot be in a place accidentally as a body is. When we say that a spirit is in a place, we mean that he contains that place by his power. He does not circumscribe a place as would a larger body in space. Rather, for an angel, to be in a place means to be effecting change1 in bodies in space. If a spirit is effecting some change on a body which is in a place, the spirit can be said to be in that place.
An Angel of the Lord descended at certain times into the pond and the water was moved. And he that went down first into the pond after the motion of the water, was made whole, of whatsoever infirmity he lay under. (John 5:4)
How does an angel come to be in a place?
An angel comes to be in a place (in the sense of containing it by his power, that is, acting on bodies which are in a place) at will. Thus, an angel does not cease to be in one place by local motion from it to another place. Rather, by a movement of the will from acting on one place to acting on another, the angel ceases to be in one place and begins to be in another.
And Habacuc said: Lord, I never saw Babylon, nor do I know the den. And the Angel of the Lord took him by the top of his head, and carried him by the hair of his head, and set him in Babylon over the den in the force of his spirit. And Habacuc cried saying: O Daniel, thou servant of God, take the dinner that God hath sent thee…. And Daniel arose and ate. And the Angel of the Lord presently set Habacuc again in his own place. (Daniel 14:35-38)
Can more than one angel be in the same place at the same time?
As we said above, for an angel to be in a place means to be effecting a change in a body in that place. The angelic power, too, is such that it effects these changes in bodies perfectly. Two men may cooperate as imperfect causes to cause a single movement in a body. For instance, to push a heavy car two or more men may cooperate, but there is a single movement in the car. This is due to an imperfection of power on the part of each man. An angel does not have this imperfection because he is of a higher order altogether. Therefore, his power to influence bodies is of a different kind not only of a different degree than the power of corporeal creatures. An angel does not effect a change matter by coming into bodily contact with it but by his will alone. Moreover, an angel does not assert his power in varying degrees. Every operation of his is perfect according to his faculties. Therefore, two angels cannot cooperate in moving a body because there cannot be two perfect (secondary) causes of the same movement.
Conclusion
To dance on the head of a pin clearly requires presence on the head of said pin. For an angel, this presence means to be effecting a change in the head of the pin itself or in a body on the head of the pin (e.g., air or other particles). The latter seems to be the more probable opinion.
If an angel were to effect a change (even an imperceptible one) in the head of the pin itself, other angels would be precluded (not as a body precludes other bodies from sharing the same place through the impenatrableness of its dimensive quantity extended in space but due to a potential confusion of causes) from effecting a change in the same body. Therefore, in that case, only one angel would be able to dance on the head of the pin.
If the head of the pin is not treated as a single body but rather as a collection of atoms or even subatomic particles, then the case is equivalent to effecting local motion in particles on the head of the pin.
If it is a question of effecting changes in the particles located on the head of the pin, then the number of angels would have an maximum equal to the number of particles present on the head of the pin. Undoubtedly, other angels could move more particles onto the head of the pin for the sake of effecting changes in them. Therefore, a maximum number of angels is established at the maximum number of particles that could be present on the head of the same pin.
Therefore, presuming strictly contemporaneous dancing (though angels could change places so quickly that we could not possibly notice), the number of angels who could dance on the head of the pin is limited to the lesser of two numbers: the maximum number of particles that could be present on the head of the pin and the difference between the total number of angels and the number of angels occupied in places other than the pin.
Or:
A = min((Maximum Number of Particles on the Head of the Pin), (Total Number of Angels - Number of Otherwise Occupied Angels))
This is the maximum number. Obviously, in practice, if an angel were to exert his power over all the particles on the head of a pin simultaneously, all other angels would be precluded from presence there.
1 An angel cannot effect every kind of change in a body since he cannot immediately give a form to matter nor effect a substantial change. He can, however, effect local motion in matter immediately because this does not cause an intrinsic change but only an extrinsic change of place in the body thus moved. An angel can indirectly cause different kinds of changes than local motion in bodies by using other bodies. “Dancing,” however, here is taken to mean effecting local motion in a body whereby the angel is said to be in one place and then another.