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Dante’s universe as 3-sphere April 22, 2008

Posted by David Pierce in Uncategorized.

The quotations about the universe as an infinite sphere remind me of another kind of sphere that is beyond our usual powers of imagination.
In the one-hundred cantos of the Divine Comedy, Dante travels from the central point of the earth out into the heavens. He passes through the spheres of

  1. the moon,
  2. Mercury,
  3. Venus,
  4. the sun,
  5. Mars,
  6. Jupiter,
  7. Saturn,
  8. the fixed stars,
  9. the Primum Mobile.

In this ninth sphere, he sees another point, surrounded by circles or spheres. These spheres are angels, named from inside to outside:

  1. Seraphim,
  2. Cherubim,
  3. Thrones,
  4. Dominions,
  5. Virtues,
  6. Powers,
  7. Principalities,
  8. Archangels,
  9. Angels.

These are named in Canto XXVIII of the Paradiso, where Dante reports (text from the Princeton Dante Project):

16 I saw a point that flashed a beam of light
17 so sharp the eye on which it burns
18 must close against its piercing brightness.
19 The star that, seen from here below, seems smallest
20 would seem a moon if put beside it,
21 as when one star is set beside another.
22 As near, perhaps, as a halo seems to be
23 when it encircles the light that colors it,
24 where the vapor that forms it is most dense,
25 there whirled about that point a ring of fire
26 so quick it would have easily outsped
27 the swiftest sphere circling the universe.
28 This point was encircled by another ring,
29 and that by the third, the third by the fourth,
30 the fourth by the fifth, and the fifth by the sixth.
31 Higher there followed the seventh, now spread so wide
32 that the messenger of Juno, in full circle,
33 would be unable to contain its size.
34 And so, too, the eighth and ninth,
35 each one revolving with diminished speed
36 the farther it was wheeling from the first.
37 And that one least removed from the blazing point of light
38 possessed the clearest flame, because, I think,
39 it was the one that is the most intruthed by it.
40 My lady, who saw me in grave doubt
41 yet eager to know and comprehend, said:
42 ‘From that point depend the heavens and all nature.
43 ‘Observe that circle nearest it,
44 and understand its motion is so swift
45 because it is spurred on by flaming love.’
46 And I to her: ‘If the universe were arranged
47 in the order I see here among these wheels
48 I would be content with what you’ve set before me.
49 ‘However, in the world of sense we see
50 the farther from the center they revolve
51 the more divinity is in their orbits.

Beatrice then explains that the innermost rings of angels correspond to the outermost heavenly spheres: so n corresponds to 10-n in the lists above.

In a lecture given in Ankara a while back, Piergiorgio Odifreddi suggested (as I recall) that the two spheres or rather balls—of the heavens, and of the angels—should be considered as identified along the 2-spheres that are their boundaries, so that a 3-sphere is obtained, with Lucifer and God as antipodal points.
I note Dante’s description of the latter Point in lines 11–12 of Canto XXX:

1 About six thousand miles away from here
2 the sixth hour burns and even now this world
3 inclines its shadow almost to a level bed,
4 when, deep in intervening air, above us,
5 begins such change that here and there,
6 at our depth, a star is lost to sight.
7 And, as that brightest handmaid of the sun advances,
8 the sky extinguishes its lights,
9 even the most beautiful, one by one.
10 Not otherwise the victory that revels
11 in eternal joy around the point that overcame me
12 and seems enclosed by that which it encloses
13 little by little faded from my sight,
14 so that, compelled by seeing nothing and by love,
15 I turned my eyes to gaze on Beatrice.

As a symbol for a world that cannot be fully comprehended, the three-dimensional surface of a four-dimensional body may serve as well as something infinite in extent.

Identity and Categorification April 21, 2008

Posted by Alexandre Borovik in Uncategorized.

A paper by  Andrei Rodin, arXiv:math/0509596 [pdf]. Appears to be relevant to our project.

A circle with the center everywhere April 3, 2008

Posted by Alexandre Borovik in Uncategorized.

A collection of quotes:

Hermes Trismegistus, “thrice-great Hermes” “God is an infinite sphere, the center of which is everywhere, the circumference nowhere.” Book of the 24 Philosophers.

Alain of Lille “God is an intelligible sphere, whose center is everywhere, and whose circumference is nowhere.”

Pascal: “The whole visible world is only an imperceptible atom in the ample bosom of nature. No idea approaches it. We may enlarge our conceptions beyond all imaginable space; we only produce atoms in comparison with the reality of things. It is an infinite sphere, the center of which is everywhere, the circumference nowhere. In short, it is the greatest sensible mark of the almighty power of God that imagination loses itself in that thought.”

Also apparently, “Let him contemplate all nature in its awful and finished magnificence; let him observe that splendid luminary, set forth as an eternal lamp to enlighten the universe; let him view the earth as a mere speck within the vast circuit described by that luminary; let him think with amazement, that this vast circuit itself is only a minute point , compared with that formed by the revolutions of the stars…All that we see in of the creation, is but an almost imperceptible streak in the vast expanse of the universe. No idea can approximate its immense extent…This is an infinite sphere, the center of which is everywhere, but its circumference nowhere. In short, it is one of the greatest sensible evidences of the almightiness of God, that our imagination is overwhelmed by these reflections.”

In “Pascal’s Sphere,” Borges’ narrator lists dozens of variations of a single image, a circle that stands alternately single image, a circle that stands alternately for God, nature, the universe, infinity. Culminating his enumeration is Pascal’s image for the universe: “an infinite sphere, the center of which is everywhere, the circumference nowhere.” Indeed, Borges himself adds to the list in his story “The Library of Babel,” the Library is described as “a sphere whose exact center is anyone of its hexagons and whose circumference is inaccessible” (Labyrinths 52).

According to Borges, Pascal hated “the universe. He was sorry the firmament could not speak; he compared our lives to those of shipwrecked men on a desert island … and he expressed his fillings [saying nature] is an infinite sphere, the center of which is everywhere, the circumference nowhere.”

[Seems to me from the quotations, especially the second, that Borges had Pascal wrong.–D.C.]