What was once just a rhetorical illustration used to demonstrate the futility of out-of-touch theological debates is now a exciting science experiment you can conduct in your very own home!

What you will need:

Instructions: Insert the pin upright into a sturdy surface, such as a pin cushion or Styrofoam block. Begin playing "The Song That Doesn't End (Extended Version)" and instruct the angels to step onto the pin and begin dancing. Count each angel, stopping only when no more angels can dance on the pin, and remembering to make sure all of the angels are dancing on the pin and not just hovering above it, so as to avoid a potential source of error. Repeat several times, removing all angels from the pin after each trial. From these trials, determine the average number of angels that can dance on the head of a pin.

• Procure a copy of Maurice Durufle's "Ubi Caritas et Amor" and repeat the experiment. Discuss the effect that music religious in nature has on your results.
• Does the type of dancing affect the number of angels who can dance? Experiment with such dancing styles as the foxtrot, the Electric Slide, and the clueless male arm flail.
• Discuss possible sources of error, such as pin imbalance, drunken angels who keep falling off of the pin's head, or angels who won't dance if they don't know the song.

Next week: Does a watched pot ever boil?

Contrary to popular belief, this question was not debated by medieval philosophers. But today with analytic philosophy, we can! (Such miracles of this modern world!)

First of all, we use Strawson's notion of presupposition. ``How many angels can dance on the head of a pin?'' might previously have been called a complex question, but now we can call it an ordinary question that presupposes the proposition ``It is possible that an angel dance on the head of a pin.'' (Not that Strawson would actually consider a question like this, of course.) So a reasonable answer to this question must presuppose this proposition to be true; an answer along the lines of ``Fool, there's no such thing as angels!'' would rightly be rejected by the questioner as irrelevant and inappropriately abusive.

This is an important point, and illustrates one of the features of philosophy not often noticed by non-philosophers: much of being a good philosopher involves being extremely sympathetic to other views, even ones which might at first seem ludicrous. This is because the philosopher should not be interested in whether a proposition is true or false, but whether a particular argument or pattern of argument is valid or invalid. (This can often lead to humility on the part of the philosopher as e realizes that e had been guilty of using fallacious arguments to emself to justify eir ordinary beliefs.) But I digress. Back to an analytic exploration of the angels-dancing-on-pins question.

We next apply Kant's method of transcendental deduction. What would the world have to be like for it to be possible for an angel to dance on the head of a pin? Well, first of all, there have to be angels, and they have to be reasonably similar to people, since they can dance. Also, if even one angel can dance on the head of a pin, angels have to be pretty small. This means that angels must be at least partially physical in nature, since what is not physical cannot have size, and is likely not to be similar to humans.

To proceed further, we must develop at least partial answers to the questions: ``How small can an angel be and still be similar to a person?'' and ``How small can an angel be and still be able to dance?''

The first question engages us with some of the most difficult questions tackled by philosophers. What is a person? What is close to being a person---that is, what qualities might an entity have that would make it more or less of a person? These questions, though interesting, unfortunately detract from the present discussion. Further, common answers to the question of person-making qualities, such as ``having consciousness'', ``dreaming'', ``having moral feelings'', or ``being able to write a sonnet on the Firth of Forth'', do not address issues related to the question. Thus it seems that an answer to the question: ``How small can an angel be and still be able to dance?'' is sufficient for our purposes.

This question, too, opens up a difficult issue. Is space continuous or discrete? If the former, it is arguable that there is no positive lower bound on the size of dancing angels. If the latter, there must be some lower bound, namely, the size of the smallest unit of space. But clearly this is too small, since dancing involves movement not only of the body but movement within the body (such as flailing arms or shifting feet). We might claim that, assuming space is discrete, the smallest possible size of a dancing angel is three times the size of the smallest unit of space, since this is the smallest volume of space that can have different shapes and therefore can be considered bendable.

A potential ambiguity in the question is the notion of the head of a pin. There is little point in considering the size of each angel if it is unknown what the size of the head of the pin is. We can stipulate that an average head of a pin is 3.5 millimeters wide, assuming that the pin being referred to is not a pushpin. The surface area of this, assuming the head is spherical, is 12.25 pi square millimeters.

At this point, the strictly philosophical analysis of the question must stop. It is still unresolved whether space is continuous or discrete, and if space is discrete, there are no obvious philosophical reasons to claim some particular size as the smallest possible. Thus, a preliminary answer to the question ``How many angels can dance on the head of a pin?'' is: If space is continuous, the number is probably unlimited; if space is discrete, then the number is finite and is a function of the surface area of the head of the pin and of the amount of surface area taken up by a being three times the size of the smallest unit of space.

This node seems to be infected with a certain modernistic smugness. Medieval philosophers could have and certainly did discuss "How many angels can dance on the head of a pin?" with the utmost seriousness.

While the pin metaphor is not used, Thomas AquinasSumma Theologiae certainly covers all the metaphysics necessary.

Aquinas was not talking about the images of fat, winged cherubim which come to mind when we hear the word “angel”. By discussing angels, Aquinas reasoned about non-corporeal substances. Today, we might similarly discuss whether it makes sense to talk about a subatomic particle which has energy, position and velocity but no mass. The Summa’s section on angels (Questions 50-64) is the first after the topic of creation. Angels are pure spirit, and the section on angels stands, in the scheme of the Summa, for a discussion of the nature of spirit. That section is followed by one treating matter, and then a section on the combination of spirit and matter, which is a Human Being.

The Summa is presented in a debate format: a question is asked, then positions pro and con and stated. The first question regarding angels is: do purely spiritual things exist, or is spirit found only in conjunction with matter? Today, one might ask, does consciousness exist apart from the brain? The referents are different but the metaphysical/philosophical question is the same.

As for the dancing, the question is, do purely spiritual beings occupy any space? Do they have extension as well as location? For a pin head, the possible answers are no angels (they have neither place nor space), one angel (they have both place and space), and infinitely many angels (they have place but take up no space).

In Aquinas’ view, the correct answer was “infinitely many”: while angels have no bodies and occupy no space, they do have location. The debate therefore illustrates the distinction between extension and location. Fundamental metaphysical questions like this were certainly debated in medieval universities, whether or not the angels and pinheads illustration was used.

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