How many angels can dance on the point of a pin? And how many fermions?

“How many angels can dance on the point of a pin?”

This question is always used as a derisive example to show how remote from reality late-mediaeval scholastic theology was. I have never seen any evidence that the question was actually ever asked: everyone just assumes it was and nobody bothers to check. Nevertheless, it seems to be rather a sensible question to ask of philosophy undergraduates. Here’s why.

An angel, for the sake of this argument, is an incorporeal being. If you aren’t happy with the idea of such things existing, you can stick an ‘If’ in front of the question: “If incorporeal beings existed, how many of them could dance on the point of a pin?”.

Here are some answers:

Infinitely many. Since an angel is an incorporeal being it doesn’t take up any space. So infinitely many of them will fit in the same space.

One of each species. Things have two whichnesses: the whichness of where they are, and the whichness of what kind of thing they are. If two things, otherwise identical, occupy exactly the same place, there is no way to tell them apart and they may as well be the same thing. Since the angels on the point of a pin are identical in the “where they are” sense, they have to be different in the “what kind of thing they are” sense, otherwise they’d be the same thing.

None at all. Angels don’t have a body so they don’t have a where. Without the whereness conferred by a material body there are only two places an immaterial being can be: everywhere, or nowhere. It can’t be somewhere in particular and not elsewhere. Being on the point of a pin means being on the point of the pin and nowhere else, so it’s not something that an angel can do.

Each of these answers provides enough material for a good half-hour’s worth of discussion in a philosophy tutorial. So far from being absurd, “How many angels can dance on the point of a pin?” is actually a rich, suggestive and educational question.

Fermions

(A fermion is a particle whose quantum-mechanical spin is an odd multiple of 1/2. All the elementary particles that make up matter are fermions, and so are many atoms).

How many fermions can dance on the point of a pin?

Here are some answers:

One of each kind. The Pauli exclusion principle says that you can’t have two identical electrons (for example) in exactly the same place.

One of each kind and state. This is a more accurate statement of what goes on with the exclusion principle. Two electrons which have opposite spins can occupy the same place: for instance, they can both be in the innermost orbital of an atom. They are not identical because their spins are opposite.

None at all. According to Heisenberg’s uncertainty principle, the more accurately you measure the position of a particle, the less accurately you know its speed and direction of motion, and the more accurately you measure its speed and direction, the less accurately you know where it actually is. If you were ever able to find out, to perfect accuracy, that a fermion was absolutely stationary, you would have no idea at all where in the universe it was. What we have here is the opposite case. If we know that a fermion is exactly at a given point at a given moment in time (in this case, the point of the pin) then we have no idea at all how fast it is going or in what direction. We have no idea, therefore, where the fermion will be in the next instant. So any fermion at the point of a pin isn’t dancing on it, it’s moving in an unknown direction at an unknown speed and almost certainly will be nowhere near the pin in the next fraction of a second.

Again, this question provides material for quite a few tutorials, testing and enlarging the student’s understanding of the quantum world.

Some other questions

I am carrying a pole 20ft long and I want to get it into my garage, which is 10ft long. How fast do I need to run? This question is addressed in §2.13 of Wolfgang Rindler’s “Essential Relativity”. The answer is just under ten million miles an hour. Rindler then sabotages his own answer by saying that if the pole shrinks to 10ft as seen by the garage and therefore fits, it is equally true that the garage shrinks to 5ft as seen by the pole, which would appear to mean that the pole doesn’t fit at all. Rindler presents some clever arguments to help the student get his head round the apparent paradoxes of the length contractions of special relativity. I haven’t tried to make the arguments my own but Essential Relativity is an authoritative textbook, so I’m sure it’s right.

I want to catch a lion but I don’t want to move. How do I do it and how long will it take me? A delightful paper on “The Mathematics of Big Game Hunting” has circulated in universities in photocopied form since it was first published in the 1940s, and I expect it is now available on the Web as well. The particular example I have taken draws on the principle that any quantum object that is somewhere has a non-zero probability of being somewhere else. (This happens even if there is a barrier in the way: this “quantum tunnelling” is the way that flash memory chips work). The greater the distance between the somewhere and the somewhere else, the smaller the probability, and the bigger the object, the smaller the probability: which is why tunnelling is only a practical proposition for electrons (the smallest particles that exist) and over microscopic distances. But the probability never goes to zero, it just becomes very, very small. So if I build a cage for my lion and wait, the lion, being a (rather large) quantum object, will sooner or later be inside my cage.

Both these questions can be laughed at just as “How many angels can dance on the point of a pin?” can be laughed at, and one day perhaps they will be; but they both stimulate the mind and deepen understanding.  So did the dancing angels.

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