How many angels can dance on the head of a pin?
The question "How many angels can dance on the head of a pin?" has been used many times as a dismissal of medieval angelology in particular, and of scholasticism in general. The phrase has been used also to criticize figures such as Duns Scotus and Thomas Aquinas, who explored the intersection between the philosophical aspects of space and the qualities attributed to angels. Another variety of the question is: "How many angels can stand on the point of a pin?"
Scholasticism used these kind of questions in dialectical reasoning to extend knowledge by inference, and to resolve contradictions. The need for rationality as complementary to faith was raised as an important point for Catholic theology at the Council of Trent. The question has also been linked to the fall of Constantinople, with the imagery of scholars debating about minutiae while the Turkish besieged the city. In modern usage, it therefore has been used as a metaphor for wasting time debating topics of no practical value, or questions whose answers hold no intellectual consequence, while more urgent concerns pile up.
The fact that certain renowned medieval scholars considered similar questions is clear; Aquinas's Summa Theologica, written c. 1270, includes discussion of several questions regarding angels such as, "Can several angels be in the same place?" However the idea that such questions had a prominent place in medieval scholarship has been debated, and it has not been proved that this particular question was ever disputed. One theory is that it is an early modern fabrication,[a] as used to discredit scholastic philosophy at a time when it still played a significant role in university education. James Franklin has raised the scholarly issue, and mentions that there is a 17th-century reference in William Chillingworth's Religion of Protestants (1637), where he accuses unnamed scholastics of debating "whether a Million of Angels may not fit upon a Needle's point?" This is earlier than a reference in the 1678 The True Intellectual System Of The Universe by Ralph Cudworth. HS Lang, author of Aristotle's Physics and its Medieval Varieties (1992), says (p. 284):
The question of how many angels can dance on the point of a needle, or the head of a pin, is often attributed to 'late medieval writers'... In point of fact, the question has never been found in this form…
Peter Harrison has suggested that the first reference to angels dancing on a needle's point occurs in an expository work by the English divine, William Sclater (1575-1626). In An exposition with notes vpon the first Epistle to the Thessalonians (1619), Sclater claimed that scholastic philosophers occupied themselves with such pointless questions as whether angels "did occupie a place; and so, whether many might be in one place at one time; and how many might sit on a Needles point; and six hundred such like needlesse points." Harrison proposes that the reason an English writer first introduced the "needle’s point" into a critique of medieval angelology is that it makes for a clever pun on "needless point".
Philosopher George MacDonald Ross has identified a close parallel in a 14th-century mystical text, the Swester Katrei. Other possibilities are that it is a surviving parody or self-parody, or a training topic in debating.
In Spanish and Portuguese, the conundrum of useless scholarly debates is linked to a similar question of whether or not angels are genderless.
Dorothy L. Sayers argued that the question was "simply a debating exercise" and that the answer "usually adjudged correct" was stated as, "Angels are pure intelligences, not material, but limited, so that they have location in space, but not extension." Sayers compares the question to that of how many people's thoughts can be concentrated upon a particular pin at the same time. She concludes that infinitely many angels can be located on the head of a pin, since they do not occupy any space there:
The practical lesson to be drawn from the argument is not to use words like "there" in a loose, unscientific way, without specifying whether you mean "located there" or "occupying space there."
In the humoristic magazine Annals of Improbable Research, Anders Sandberg has presented a calculation based on theories of information physics and quantum gravity, establishing an upper bound of 8.6766×1049 angels.
In the fifth season of the science-fiction series Babylon 5, the recurring character Byron Gordon, in a conversation about a rebellion among Human Telepaths against a despotic government both asked and answered the question with a confident but cryptic "As many as are needed." Thus suggesting the specific number of angels is irrelevant, it is the existence of angels, (and by way of analogy the Telepaths and allies that follow of the message of freedom and peace against tyranny) that is important.
In the satirical novel Good Omens by Neil Gaiman and Terry Pratchett, the angel Aziraphale is said to be the only angel who could dance on the head of a pin, as he learned the gavotte in the 19th century. Also in Carpe Jugulum by Terry Pratchett Granny Weatherwax says the answer is 16 if it's an ordinary house pin.
In other contexts
Comparing mediæval superstition and modern science, George Bernard Shaw wrote in the introduction to the play Saint Joan that "The medieval doctors of divinity who did not pretend to settle how many angels could dance on the point of a needle cut a very poor figure as far as romantic credulity is concerned beside the modern physicists who have settled to the billionth of a millimetre every movement and position in the dance of the electrons." 
- Argumentation theory
- Balloon debate
- Philosophy of copyright, for a modern take of a scholastic problem
- Summa, New advent.
- "St. Thomas does not discuss the question "How many angels can dance on the point of a needle?" He reminds us that we must not think of angels as if they were corporeal, and that, for an angel, it makes no difference whether the sphere of his activity be the point of a needle or a continent (Q. lii, a.2)." (Kennedy, D. J., "Thomism", in the Catholic Encyclopedia)
- Hirsch, E. D. Jr.; Kett, Joseph F.; Trefil, James, eds. (2002). The New Dictionary of Cultural Literacy (Third ed.). Houghton Mifflin Co. Archived from the original on 3 July 2003.
- Terence McLaughlin; Joseph O'Keefe. The Contemporary Catholic School: Context, Identity And Diversity. Routledge, 2003.
- "How many angels can dance on the head of a pin?". Today's Zaman.
- Ramírez, José A. (1975). Las Andanzas Del Diablo: Confidencias de un Abogado Ingenuo. Editorial Planeta. p. 58. ISBN 9788432053375. "...reminded of the stupid story of the Byzantine empire. Everybody knows the idiotic and sometimes bloody discussions in that Empire on the sex of angels, about how many could perch at the same time on the head of a pin"
- Van Asselt, Willem J (2011). Introduction to Reformed Scholasticism. p. 65.
- Franklin 1993 p. 127.
- Peter Harrison, "Angels on Pinheads and Needles’ Points", Notes and Queries, 63 (2016), 45-47.
- G. MacDonald Ross, Angels in: Philosophy, vol. 60, 1985, pp. 499–515.
- Sayers, Dorothy L. "The Lost Tools of Learning". Gbt.org. Retrieved 14 November 2012.
- "Quantum Gravity Treatment of the Angel Density Problem". Annals of Improbable Research. 2001. Retrieved 10 May 2013.
- "Saturday Morning Breakfast Cereal". Smbc-comics.com. Retrieved 14 November 2012.
- "Saint Joan – A Chronicle Play in Six Scenes and an Epilogue". Retrieved 22 July 2015.
- Jeremy Hunsinger; Lisbeth Klastrup; Matthew M. Allen, International Handbook of Internet Research, Springer Science & Business Media
- Rutgers computer & technology law journal, Volume 20, Page 110. 1994.
- National Journal, Volume 28, Pages 1-452. Government Research Corporation. National Journal Group Incorporated, 1996
- Franklin, J., "Heads of Pins" in: Australian Mathematical Society Gazette, vol. 20, n. 4, 1993.
- Harrison, Peter. "Angels on Pinheads and Needles’ Points", Notes and Queries, 63 (2016), 45-47.
- Howard, Philip (1983), Words Fail Me, summary of correspondence in The Times on the matter
- Kennedy, D. J., "Thomism", in the Catholic Encyclopedia
- Koetsier, T. & Bergmans, L. (eds.), Mathematics and the Divine: a historical study, Ch. 14 by Edith Sylla (review)