{"id":296,"date":"2006-07-31T10:27:20","date_gmt":"2006-07-31T14:27:20","guid":{"rendered":""},"modified":"2007-04-04T14:19:14","modified_gmt":"2007-04-04T19:19:14","slug":"how-many-angels-can-dance-on-the-head-of-a-pin","status":"publish","type":"post","link":"https:\/\/biblearchive.com\/blog\/how-many-angels-can-dance-on-the-head-of-a-pin\/","title":{"rendered":"How Many Angels Can Dance on the Head of A Pin"},"content":{"rendered":"

We Christians like to get into debates that there really is
\nno answer to. And we get into a huge huff about it too. One old debate was the
\none about Angels and Pin points (not pin heads as is often cited although it
\ndoes result in a certain amount of pin-headishness). I found a few articles:
\none that confirmed that there was some wacky discussions<\/a> during the medieval
\nperiod (though this one may not have been one of them), one that offers a
\nmethodology to answer the question<\/a> and one (here’s the link and it’s also in
\nthe read more) that actually mathematically counted how many Angels can dance
\non the point of a pin<\/a>.<\/p>\n

<\/p>\n

\n

Quantum Gravity Treatment of the Angel Density Problem<\/h2>\n

by Anders
\nSandberg<\/a>
\nSANS\/NADA, Royal Institute of
\nTechnology<\/a>, Stockholm, Sweden<\/p>\n

[EDITOR’S NOTE: we apologize for the lack of clear formatting,in this
\nweb version, of the mathematical formulae.]<\/em><\/p>\n

Abstract<\/h4>\n

We derive upper bounds for the density of angels dancing on the point of
\na pin. It is dependent on the assumed mass of the angels, with a maximum
\nnumber of 8.6766*10exp49 angels at the critical angel mass (3.8807*10exp-34
\nkg).<\/p>\n

Ancient Question, Modern Physics<\/h4>\n

“How many angels can dance on the head of a pin?” has been a
\nmajor theological question since<\/a>
\nthe Middle Ages.[5]<\/p>\n

According to Thomas Aquinas, it is impossible for two distinct causes to
\neach be the immediate cause of one and the same thing. An angel is a good
\nexample of such a cause. Thus two angels cannot occupy the same space.[2]
\nThis can be seen as an early statement of the Pauli
\nexclusion principle<\/a>. (The Pauli exclusion principle is a pillar of modern
\nphysics. It was first stated in the twentieth century, by Pauli.)<\/p>\n

However, this does not place any upper bound on the density of angels in
\na small area, because the size r of angels remains undefined and could possibly
\nbe arbitrarily small. There have also been theological criticisms of any
\nassumption of angels as complete causes.<\/p>\n

Stating the Question Correctly<\/h4>\n

The basic issue is the maximal density of active angels in a small volume.
\nIt should be noted that the original formulation of the problem did not
\nrefer to the head of a pin (R\u00c2\u00bc1 mm) but to the point of the pin. Therefore,
\nthe point, not the head, of the pin is the region that will be studied in
\nthis paper.<\/p>\n

One of the first reported attempts at a quantum gravity treatment of the
\nangel density problem that also included the correct end of the pin was
\nmade by Dr.
\nPhil Schewe<\/a>. He suggested that due to quantum gravity space is likely
\nnot infinitely divisible beyond the Planck length scale of 10exp-35 meters.
\nHence, assuming the point of the pin to be one \u00c3\u2026ngstr\u00c3\u00b6m across
\n(the size of a scanning tunnelling microscope tip) this would produce a
\nmaximal number of angels on the order of 1050 since they would not have
\nmore places to fill.[1]<\/p>\n

While this approach does produce an upper bound on the possible density
\nof angels, it is based on the Thomist assumption of non-overlap.<\/p>\n

Since angels can be presumed to obey quantum rules when packed at quantum
\ngravity densities, the uncertainty relation will cause their wave functions
\nto overlap significantly even if there is a strong degeneracy pressure.
\nIf the non-overlap assumption is relaxed, this approach cannot derive an
\nupper bound.<\/p>\n

Quantum
\nGravitational<\/a> Treatment<\/h4>\n

A stricter bound based on information physics can be derived that is not
\nbased on overlap assumptions, but merely the localisation of angelic information.<\/p>\n

Assuming that each angel contains at least one bit of information (fallen
\n\/ not fallen), and that the point of the pin is a sphere of diameter of
\nan \u00c3\u2026ngstr\u00c3\u00b6m (R=10exp-10 m) and has a total mass of M=9.5*10exp-29
\nkilograms (equivalent to that of one iron atom), we can use the Bekenstein
\nbound<\/a>[3] on information to calculate an upper bound on the angel density.
\nIn a system of diameter D and mass M, less than kDM distinguishable bits
\ncan exist, where k=2.57686*10exp43 bits\/meter kg.[7] This gives us a bound
\nof just 2.448*10exp5 angels, far below the Schewe bound.<\/p>\n

Note that this does not take the mass of angels into account. A finite
\nangel mass-energy would increase the possible information density significantly.
\nIf each angel has a mass m, then the Bekenstein bound gives us N<kD(M+Nm).
\nBeyond mcrit>1\/kD \u00c2\u00bc3.8807*10exp-34 kg this produces an unbounded maximal
\nangel density as each angel contributes enough mass-energy to allow the
\ninformation of an extra angel to move in, and so on.<\/p>\n

However, if angels have mass, then the point of the pin will collapse into
\na black hole if c2R\/2G< Nm (here I ignore the mass of the iron atom at
\nthe tip).4 For angels of human weight (80 kg), we get a limit of 4.2089*10exp14
\nangels. The maximal mass of any angel amenable to dance on the pin is 3.3671*10exp16
\nkg; at this point there is only room for a single angel.<\/p>\n

The picture that emerges is that, for low angel masses, the number is bounded
\nby the Bekenstein bound, and increases hyperbolically as mcrit is approached.
\nHowever, the black hole bound decreases and the two bounds cross at mmax=1\/(4GkM\/c<\/em>exp2+kD),
\nvery slightly below mcrit. This corresponds to the maximal angel density
\nof Nmax=8.6766*10exp49 angels (see figure).<\/p>\n

<\/p>\n

Maximum number of angels for a given mass. The allowed
\nregion is bounded from above by the line c2R\/2G=Nm (gravitational collapse)
\nand the curve N=kD(M+Nm) (information density) which has an asymptote
\nfor mcrit, and from below by N=0. The maximal number of angels occurs
\nat the intersection of the gravitational bound and the asymptote at mcrit<\/p>\n

Dance Dynamics<\/h4>\n

If the angels dance very quickly and in the same direction, then the angular
\nmomentum could lead to a situation like the extremal Kerr metric, where
\nno event horizon forms (this could also be achieved by charging the angels).[4]
\nHence the number of dancing angels that can crowd together is likely much
\nhigher than the number of stationary angels.<\/p>\n

However, at these speeds the friction caused by their interaction with
\nthe pin is likely to vaporise it or at least break it apart. Even for a
\nmodest speed of 1 m\/s the total kinetic energy of Nmax angels of mass mcrit
\nwould be 1.682*10exp16 J. In the case of charged angels at relativistic
\ndensities, pair-creation in their vicinity would likely cause the charge
\nto dissipate over time,6 and charge transfer to the pin would also likely
\ninduce electromechanical forces beyond any material tolerances.<\/p>\n

The uncertainty relation also imposes a limitation on the dance. Since
\nthe uncertainty in position of the angels by assumption is less than the
\nsize of the point \u00c3\u0090x?R we find that the uncertainty in momentum must be
\n\u00c3\u0090p?hbar\/R, and this leads to a velocity uncertainty \u00c3\u0090v>hbar\/Rm. If m=
\nmcrit we get \u00c3\u0090v>> 8.6766*10exp59 m\/s (>> c), which shows that:<\/p>\n

(1) the angels must dance with speeds near the velocity of light in order
\nto obey quantum mechanics;<\/p>\n

(2) a full relativistic treatment is necessary; and<\/p>\n

(3) that the precision of the dance must break down due to quantum effects.
\nThis can be used to rule out certain types of dance due to their high precision
\nrequirements.<\/p>\n

Discussion<\/h4>\n

We have derived quantum gravity bounds on the number of angels that can
\ndance on the tip of a needle as a function of the mass of the angels. The
\nmaximal number of angels — 8.6766*10exp49 — is achieved near the critical
\nmass mcrit>1\/kD \u00c2\u00bc3.8807*10-34 kg, corresponding to the transition from
\nthe information-limited to the mass-limited regime. It is interesting to
\nnote that this is of the same order of magnitude as the Schewe bound.<\/p>\n

Angel physics has until now mainly employed theological methods, but as
\nthis paper shows, modern information physics, quantum gravity and relativity
\ntheory provide powerful tools for exploring the dynamics and statics of
\nangels.<\/p>\n

These bounds are only upper bounds, and do not take into account the effects
\nof a finite number of available angels, degeneracy pressures if angels obey
\nthe Pauli exclusion principle as suggested by Aquinas, or the theo-psychology
\nof the angels themselves. The exact dance dynamics also clearly play a major
\nrole. A full relativistic treatment of the dance appears as a promising
\navenue for further tightening of the bounds.<\/p>\n

Bibliography<\/h4>\n

1. Reported in the New York Times<\/em>, November 11, 1997, “Science
\nQ \\& A: Dancing Angels,” by C. Claiborne Ray <http:\/\/www.nytimes.com\/learning\/students\/scienceqa\/archive\/971111.html<\/a>>.
\nDr. Phil Schewe presented his idea at a meeting of the Society for Literature
\nand Science in 1995. It should be noted that the given density in the New
\nYork Times is wrong, as it did not count the area, just the length.
\n2. T. Aquinas, Summa Theologiae<\/em>, vol. 52, no. 3, 1266.
\n3. J. Bekenstein, Physical Review D<\/em>, vol. 23, no. 287, 1981.
\n4. Gravitation<\/em>, C. Misner, K. Thorne, and J. Wheeler, W.H. Freeman
\nand Co., 1973.
\n5. Memoirs of the Extraordinary Life, Works and Discoveries of Martinus
\nScriblerus<\/em>, 1741, A. Pope, J. Swift, J. Gay, T. Parnell, and J. Arbuthnot,
\ned. by C. Kerby-Miller, 1950; republished 1966.
\n6. “On the Dyadosphere of Black Holes,” R. Ruffini, in Proceedings
\nof Yamada Conference, Kyoto, Japan, April 1998<\/em>. <http:\/\/xxx.lanl.gov\/abs\/astro-ph\/9811232<\/a>>.
\n7. The Physics of Immortality<\/em>, Frank Tipler,. Macmillan, London,
\n1994.<\/p>\n

\u00c2\u00a9 Copyright 2001 Annals of Improbable
\nResearch (AIR)<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

We Christians like to get into debates that there really is no answer to. And we get into a huge huff about it too. One old debate was the one about Angels and Pin points (not pin heads as is often cited although it does result in a certain amount of pin-headishness). 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