Below I post a collection of thoughts on this matter, though I have not troubled to write a formal essay.
Notes from Wikipedia:
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?"
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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?"
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Philosopher and historian 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 on 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."
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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.
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And from the Cambridge Descartes Lexicon [ 6], we learn:
In the Third Meditation, Descartes introduces the idea of angel as an instance of an idea the human mind can form from other ideas, namely from the idea of human being and from that of God (AT VII 43, CSM II 29; AT VII 138–39, CSM II 99). In the Conversation with Burman, Descartes reportedly asserts that he cannot say anything about angels’ nature, except that like human minds they are thinking substances (AT V 157–58). However, in discussing Regius's theory that man is a substance by accident (per accidens), Descartes provides one counterfactual detail concerning angelic knowledge. The human mind is really and substantially united to the extension of his body. In consequence of this kind of union, the human mind perceives the mechanical modifications of his body as obscure sensations and passions. By contrast, if an angel were joined to a body, his mind would perceive the modifications of this body as they really are, that is, as the mechanical effects of the external bodies that cause them. Consequently, he would know the former only through clear and distinct ideas (AT III 493, CSMK 206) (see clarity and distinctness). An angel united to a body would realize the kind of union that a pilot bears to his ship, which Descartes opposes to the kind of union that holds in the case of human nature (AT VII 81, CSM II 56). To Burman, Descartes adds that he deems wholly irrelevant what views Aquinas – who had been so concerned with angels to deserve to be called “Doctor Angelicus” – held concerning the nature and mind of angels.
Despite this dismissive attitude, Descartes draws on what Aquinas and John Duns Scotus say about angels in his account of the human mind and the nature of human knowledge. For example, the following Cartesian claims on the angelic mind can be traced directly to Aquinas:
1.The mind directly knows itself; its nature consists in thinking and hence it always thinks (AT III 478, CSMK 203; AT V 193, CSMK 355 (cf. Summa Theologica, I, q.56, a.1; q.58, a.1).
2.Understanding and will, and not imagination and sensation, pertain to the nature of the mind (AT VII 73, CSM II 51) (cf. Summa Theologica I, q.54, a.5).
Blogger Conant remarks:
1. Aquinas and others are speaking of some noumenal (non-phenomenal or spirit) world "intersecting" the phenomenal world of Newtonian (or even Einsteinian) space and time. (Obviously, we have used anachronisms.)
Dorothy Sayers makes a good point that immaterial intelligences are not subject to the limits of our material world.
2. Recall how Jesus and the disciples were instantly translated across the Sea of Galilee. Space and time vanish when subjected to spiritual power. Recall that the resurrected Jesus would just show up somehow to speak with disciples. He was not an angel in the sense of never having been born of a woman, but otherwise had the attributes of an angel (in fact, a "son of God" carries the same meaning as "angel" or personal representative). So we see that non-fallen angels are not subject to earthly time and space.
Because of the fact that our fallen world is to a great extent a matter of (compelled) perception, I would say that when an angel appears, that being is projecting its spiritual essence into a person's mind. This may occur "in a dream" with such lucidness that the person takes it as having "really happened."
3. An interesting question is whether humans actually occupy positions in space and time; or is that belief an artifact of human perception mechanisms?
4. Many have heard of Enrico Fermi's famous quip about space aliens from other planets: "Where are all the tourists?" Some make the same jest about angels and other "immaterial" spirits. Yet, is it not so that such beings sometimes pass unnoticed? As the Bible says Be not forgetful to entertain strangers: for thereby some have entertained angels unawares. -- Hebrews 13.2 (KJV/Authorized). And, if an angel projects itself into a human mind for a short time, certainly that being can limit its projection to one or a few persons.
5. For centuries, the image of counting angels on a pin head has been used by philosophers to scorn Aquinas and scholasticism. Yet, we see that the question performed a sound philosophical function of encapsulating serious questions concerning space, time and God's universe. Ridicule is an effective weapon, because it appeals to human pride in assuming one knows what is "obviously" so.
6. The logician Rudy Rucker observed that the "famous puzzle of how many angels can dance on the head of a pin can be viewed as a question about the relationship between the infinite Creator and the finite world. The crux of the problem is that, on the one hand, it would seem that since God is infinitely powerful, he should be able to bid an infinite number of angels to dance on the head of a pin; on the other hand, it was believed by medieval thinkers that no actually infinite collection could ever arise in the created world." [1]
But their "proofs that infinity is somehow a self-contradictory notion were all flawed," Rucker adds.
7. Yet the actual infinite was necessary if the mathematical notion of infinitesimal were to survive. Even if we replace infinitesimals with Weirstrasse's epsilon-delta limit points, the limit point is indistinct from the infinitesimal. Weirstrasse introduced Aristotle's potential infinity (a ridiculous term as a quantity or set is either finite or not) as a means of escaping infinitesimals. But if the limit points are only fictions, then space is discontinuous and there are no smooth curves. An object does not follow a smooth trajectory. Today, we can argue from quantum theory that this indeed holds, but it is doubtful one can get rid of all infinitesimals.
An infinitesimal implies a subspace in which tiny geometric relations can exist without being visualizable (much like quantum "phenomena").
Rucker points out that "Bishop Berkeley found it curious that mathematicians could swallow the Newton-Leibniz theory of infinitesimals, yet balk at the peculiarities of orthodox Christian doctrine." [1]
8. In fact, Amir Alexander has told of a tremendous cultural and ideological war against the infinitesimal led by the Society of Jesus and backed by some authoritarian Protestants. The concern was that if the world were composed of infinitely small bits, the conservative theological doctrines that underpinned the status quo ante would come apart [2]. Supposedly the worry was that Creation reflects a divine order, which the Jesuits interpreted as a "rational" pyramidal orderliness composed of "real" finite blocks in the earthly realm. Their society, founded by an aristocrat and backed by the aristocracy, was a rigorous hierarchical structure that aristocrats believed was natural. The Roman Catholic Church, with the Pope as the chief, should be hierarchical. But infinitely small quantities, to the Jesuits, were subversive of the spiritually proper order. Infinitesimals would raise doubts about reality and perhaps about God (and there is some truth in that fear). [3][4]
9. In his professional biography, Rudolph Carnap scorns the University of Chicago Philosophy Department for hearing a Ph.D dissertation on the ontological proof of the existence of God, saying he felt as though he had found himself back in the days when philosophers argued about how many angels could fit on a pinhead. Carnap laughed at a colleague's suggestion that one should remain open-minded [5]. Carnap spent much of his career trying to find a way to justify a scientific approach to philosophy, working to expunge metaphysics as a serious subdiscipline. Materialism (at times under another name) without Marxism was his aim. There is no consensus that he succeeded.
1. Infinity and the Mind -- the science and philosophy of the infinite by Rudy Rucker (Princeton 1995).
2. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (Scientific American 2014).
3. A mixed review of Alexander's Infinitesimal
http://www.historytoday.com/reviews/infinitesimal
In response to one criticism, I would say an indivisible may not be quite the same as an infinitesimal. And yet if we regard the reals as continuous points on a finite line segment, then we could regard each real as an infinitesimal. The Leibnizian type of infinitesimal which shows up in calculus is regarded as an infinitely small non-zero quantity with 0 dimension. In principle, a real number or a Leibnizian tangent point are seemingly equally goofy.
4. Stanford Encyclopedia on infinitesimal, indivisible
https://plato.stanford.edu/entries/continuity/
5. The Philosophy of Rudolph Carnap, Paul Arthur Schilpp ed. (Open Court 1963).
*****
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?"
*****
Philosopher and historian 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 on 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."
*****
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.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
And from the Cambridge Descartes Lexicon [ 6], we learn:
In the Third Meditation, Descartes introduces the idea of angel as an instance of an idea the human mind can form from other ideas, namely from the idea of human being and from that of God (AT VII 43, CSM II 29; AT VII 138–39, CSM II 99). In the Conversation with Burman, Descartes reportedly asserts that he cannot say anything about angels’ nature, except that like human minds they are thinking substances (AT V 157–58). However, in discussing Regius's theory that man is a substance by accident (per accidens), Descartes provides one counterfactual detail concerning angelic knowledge. The human mind is really and substantially united to the extension of his body. In consequence of this kind of union, the human mind perceives the mechanical modifications of his body as obscure sensations and passions. By contrast, if an angel were joined to a body, his mind would perceive the modifications of this body as they really are, that is, as the mechanical effects of the external bodies that cause them. Consequently, he would know the former only through clear and distinct ideas (AT III 493, CSMK 206) (see clarity and distinctness). An angel united to a body would realize the kind of union that a pilot bears to his ship, which Descartes opposes to the kind of union that holds in the case of human nature (AT VII 81, CSM II 56). To Burman, Descartes adds that he deems wholly irrelevant what views Aquinas – who had been so concerned with angels to deserve to be called “Doctor Angelicus” – held concerning the nature and mind of angels.
Despite this dismissive attitude, Descartes draws on what Aquinas and John Duns Scotus say about angels in his account of the human mind and the nature of human knowledge. For example, the following Cartesian claims on the angelic mind can be traced directly to Aquinas:
1.The mind directly knows itself; its nature consists in thinking and hence it always thinks (AT III 478, CSMK 203; AT V 193, CSMK 355 (cf. Summa Theologica, I, q.56, a.1; q.58, a.1).
2.Understanding and will, and not imagination and sensation, pertain to the nature of the mind (AT VII 73, CSM II 51) (cf. Summa Theologica I, q.54, a.5).
Blogger Conant remarks:
1. Aquinas and others are speaking of some noumenal (non-phenomenal or spirit) world "intersecting" the phenomenal world of Newtonian (or even Einsteinian) space and time. (Obviously, we have used anachronisms.)
Dorothy Sayers makes a good point that immaterial intelligences are not subject to the limits of our material world.
2. Recall how Jesus and the disciples were instantly translated across the Sea of Galilee. Space and time vanish when subjected to spiritual power. Recall that the resurrected Jesus would just show up somehow to speak with disciples. He was not an angel in the sense of never having been born of a woman, but otherwise had the attributes of an angel (in fact, a "son of God" carries the same meaning as "angel" or personal representative). So we see that non-fallen angels are not subject to earthly time and space.
Because of the fact that our fallen world is to a great extent a matter of (compelled) perception, I would say that when an angel appears, that being is projecting its spiritual essence into a person's mind. This may occur "in a dream" with such lucidness that the person takes it as having "really happened."
3. An interesting question is whether humans actually occupy positions in space and time; or is that belief an artifact of human perception mechanisms?
4. Many have heard of Enrico Fermi's famous quip about space aliens from other planets: "Where are all the tourists?" Some make the same jest about angels and other "immaterial" spirits. Yet, is it not so that such beings sometimes pass unnoticed? As the Bible says Be not forgetful to entertain strangers: for thereby some have entertained angels unawares. -- Hebrews 13.2 (KJV/Authorized). And, if an angel projects itself into a human mind for a short time, certainly that being can limit its projection to one or a few persons.
5. For centuries, the image of counting angels on a pin head has been used by philosophers to scorn Aquinas and scholasticism. Yet, we see that the question performed a sound philosophical function of encapsulating serious questions concerning space, time and God's universe. Ridicule is an effective weapon, because it appeals to human pride in assuming one knows what is "obviously" so.
6. The logician Rudy Rucker observed that the "famous puzzle of how many angels can dance on the head of a pin can be viewed as a question about the relationship between the infinite Creator and the finite world. The crux of the problem is that, on the one hand, it would seem that since God is infinitely powerful, he should be able to bid an infinite number of angels to dance on the head of a pin; on the other hand, it was believed by medieval thinkers that no actually infinite collection could ever arise in the created world." [1]
But their "proofs that infinity is somehow a self-contradictory notion were all flawed," Rucker adds.
7. Yet the actual infinite was necessary if the mathematical notion of infinitesimal were to survive. Even if we replace infinitesimals with Weirstrasse's epsilon-delta limit points, the limit point is indistinct from the infinitesimal. Weirstrasse introduced Aristotle's potential infinity (a ridiculous term as a quantity or set is either finite or not) as a means of escaping infinitesimals. But if the limit points are only fictions, then space is discontinuous and there are no smooth curves. An object does not follow a smooth trajectory. Today, we can argue from quantum theory that this indeed holds, but it is doubtful one can get rid of all infinitesimals.
An infinitesimal implies a subspace in which tiny geometric relations can exist without being visualizable (much like quantum "phenomena").
Rucker points out that "Bishop Berkeley found it curious that mathematicians could swallow the Newton-Leibniz theory of infinitesimals, yet balk at the peculiarities of orthodox Christian doctrine." [1]
8. In fact, Amir Alexander has told of a tremendous cultural and ideological war against the infinitesimal led by the Society of Jesus and backed by some authoritarian Protestants. The concern was that if the world were composed of infinitely small bits, the conservative theological doctrines that underpinned the status quo ante would come apart [2]. Supposedly the worry was that Creation reflects a divine order, which the Jesuits interpreted as a "rational" pyramidal orderliness composed of "real" finite blocks in the earthly realm. Their society, founded by an aristocrat and backed by the aristocracy, was a rigorous hierarchical structure that aristocrats believed was natural. The Roman Catholic Church, with the Pope as the chief, should be hierarchical. But infinitely small quantities, to the Jesuits, were subversive of the spiritually proper order. Infinitesimals would raise doubts about reality and perhaps about God (and there is some truth in that fear). [3][4]
9. In his professional biography, Rudolph Carnap scorns the University of Chicago Philosophy Department for hearing a Ph.D dissertation on the ontological proof of the existence of God, saying he felt as though he had found himself back in the days when philosophers argued about how many angels could fit on a pinhead. Carnap laughed at a colleague's suggestion that one should remain open-minded [5]. Carnap spent much of his career trying to find a way to justify a scientific approach to philosophy, working to expunge metaphysics as a serious subdiscipline. Materialism (at times under another name) without Marxism was his aim. There is no consensus that he succeeded.
1. Infinity and the Mind -- the science and philosophy of the infinite by Rudy Rucker (Princeton 1995).
2. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (Scientific American 2014).
3. A mixed review of Alexander's Infinitesimal
http://www.historytoday.com/reviews/infinitesimal
In response to one criticism, I would say an indivisible may not be quite the same as an infinitesimal. And yet if we regard the reals as continuous points on a finite line segment, then we could regard each real as an infinitesimal. The Leibnizian type of infinitesimal which shows up in calculus is regarded as an infinitely small non-zero quantity with 0 dimension. In principle, a real number or a Leibnizian tangent point are seemingly equally goofy.
4. Stanford Encyclopedia on infinitesimal, indivisible
https://plato.stanford.edu/entries/continuity/
5. The Philosophy of Rudolph Carnap, Paul Arthur Schilpp ed. (Open Court 1963).
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6. The Cambridge Descartes Lexicon by Emanuela Scribano, Università Ca' Foscari Venezia
Edited by Lawrence Nolan, California State University, Long Beach
Publisher: Cambridge University Press
Publisher: Cambridge University Press
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