Slides

Record the rhythm of the heavens so that you can adjust your life
and plans and expectations to them.
1
2
In about 2500 B.C., the star Sirius became visible in the morning
twilight each year a day or so before the annual flooding of the Nile
began at the capitol of Memphis. This flooding was source of
Egypt's life and power.
The rays of the Sun (symbol of Ra) end little hands that caress the
Pharaoh.
3
4
By examination of long collected records of the skies, Babylonians
were able to discover celestial cycles, and even predict such celestial
events as eclipses and planetary positions. They produced empirical
rules and algorithms which were sufficient for astrological
predictions to get an idea of the wishes of the gods (astral religion).
Although these were useful to perceive and predict the will of the
gods through astrology, the rules did not give UNDERSTANDING.
In the tympanium above the right doorway of the west portal of
Chartres Cathedral, the Virgin is shown enthroned. On the
surrounding archivolt are represented the muses of the 7 liberal arts:
- lower trivium (grammar, logic, rhetoric) and the higher quadrivium
(arithmetic, geometry, astronomy, music)
- also portrayed is the ancient philosopher that, during the middle
ages, was associated with a given liberal art.
- for example, Euclid appears next to Geometry
5
6
- Who is associated with Music? Surprising answer:
PYTHAGORAS !
a) Plato believed the real world was eternal and perfect. This real
world was perceived through the mind.
- Discovered the mathematical basis of music harmony (describe)
b) Sensible world was shadow of real world. Plato points up,
indicating that the heavens was closest to true reality that we can see.
(Aside: Aristotle gestures downward indicating "This world we see
is real".)
-Pointed out it was independent of material (brass, wood, gut):
Music is in the mathematics, not the material.
-Note that 4/7 of the traditional liberal arts (the quadrivium) are
mathematical.
c) Mathematics is the purest form of thought. Gives absolute truth.
Above Plato's Academy was written:
Let no one ignorant of mathematics enter
7
8
a) Stars look like they are on spheres undergoing UNIFORM
CIRCULAR MOTION
b) Unchanging realm. Creator of Universe would have made it
perfect. Why change?
c) Uniform circular changes but is changeless. No speed up or slow
down.
9
10
a) Planetary motions (Greek for wanderer) are more complicated.
(Give hand description)
b) Eudoxos tried to explain it using the Axiom of Astronomy:
Uniform circular motion
The heavenly motions are uniform and circular, or composed of
uniform and circular parts
11
12
a) Greatest ancient astronomer, wanted accuracy but did not require
physical "understanding".
b) His system was more complicated, circles upon circles that did
not undergo uniform circular motion.
c) But his system was NOT thought of as "real" just a mathematical
technique to produce accurate predictions.
d) His chief work was titled Syntaxis Mathematika (The
Mathematical Construction). Several centuries later, the Islamic
culture titled Ptolemy's work the Almagest (the Greatest)
e) Split between astronomy (accurate predictions) and physics
(underlying reality)
13
14
St. Augustine (354 - 430) - Botticelli fresco in Church of
Ognissanti (Florence, 1480)
-Had a classical education. Came under the sway of
Neoplatonism and Neopythagoreanism.
- For a time, Augustine had believed that rational thought could lead
to truth, and knowledge of God.
- But then took developing line that the only things worth knowing
are those conducive to loving God.
- Attention to this corruptible natural world (i.e., natural philosophy)
does not fit into this.
15
16
- Augustine's rejection of this world much influenced by the
crumbling of the Roman Empire (sack of Rome itself by Alaric in
410) and beginning of the Dark Ages. Concentrate on the next
world.
Men flagellating themselves
a)
Mortification of the flesh. Think of the next world.
Reject attention to this world (e.g., science)
b) Refer to scene in the film The Seventh Seal.
- Mortification of the flesh. Reject attention to this world (e.g.,
science
c) But the Crusades of the 11th & 12th centuries exposed the
Western Christians to the advanced civilizations of
Byzantium and Islam, which preserved & advanced the
knowledge and philosophy of the ancient Greek
civilization.
d) Crusaders brought back new luxuries and new ideas
(which were often fought – edict of 1215 forbade the
teaching of Aristotelian at University of Paris).
17
18
The Triumph of St. Thomas Aquinas (1225-74) - in St Marie
Novella (Florence, 14th century)
a) In his Summa Theologica, he joined Christian dogma to
Aristotelian philosophy and science
b) Reason, along with Faith, was path to truth.
c) In the fresco:
•Above Aquinas are 4 cardinal virtues (Prudence, Temperance,
Fortitude, Justice)
•In his hands, he displays Wisdom of Soloman: "I prayed and
understanding was given me".
•Averroes, Sabellus and Arius at his feet. (Left John the
Evangelist & right Moses with rays)
•Bottom row of women are 7 liberal arts; Theologica middle-left.
•the 4 on the right are mathematical arts: Arithmetic, Geometery,
Astronomy, Music;
•the men in front are founders of each art.
19
20
Old Testament miniature (French 13c). Ostereich Natbiblio, Vienna,
Cod 255a (or Cod 2554), fol. 4
(Nuremburg Chronicle, 1493)
Revolving spheres with God on high, looking down and caring for
the corruptible earth at the center. Cozy Aristotelian Christian
theology. But there was still a strong neoplatonic spirit.
- Early neoplatonic Christianity fits Christian notion of God's
perfection.
21
22
Marsilio Ficino (1433-1499), the great Renaissance poet, scholar
and philosopher, wrote about the perfection of Light (Lux), and the
identification of the sun with it:
- Imbued with Renaissance learning and attitudes
-Disturbed by some discrepancies between observed planetary
positions, and the predictions based on Ptolemaic methods. But was
also disturbed because of his neoplatonic and neopythagorean
attitudes, which show through,
- Wanted to get back to axiom of astronomy.
- He was also affected by the Renaissance view of light: the Good
should be at the center.
23
24
- Copernicus put Sun at center in his De Revolutionibus Orbium
Caelestium (Nuremberg, 1543).
Listen to neoplatonic Ficino speaking through Copernicus:
25
26
But if Ptolemy had to introduce non-uniform circular motion to get
accuracy, Copernicus could not get by so easily. He had to add little
circles on bigger circles to get accuracy, but cleverly used uniform
circular motion. This "revisionist" aspect of the Copernican system
is rarely shown to the public.
Believed Copernicus hypothesis. Sun should be at center. Seemed
better, more appropriate order.
Became assistant of Tycho Brahe, the Imperial Mathematician of
Holy Roman Emperor Rudolph II
- Tycho died in 1601 of complications from a burst bladder after
drinking bout with the Emperor, from which he could not relieve
himself. (Modern scholarship suggests he may have died,
instead, of mercury poisoning. Foul play cannot be ruled out but
the favored theory is that he took medicine of high mercury
content - common in those days - to treat a longstanding urinary
problem.)
Rheinhold’s annotations in the margins praise Copernicus for this.
Very little commentary on cosmology for that was not really taken as
“real” – for religious and philosophical authority said the ear held the
central position in the cosmos.
Astronomers respected Copernicus for his relatively good accuracy
and his aesthetic return to the Axiom of Astronomy. Copernicus may
be thought of as a counter-revolutionary in some ways.
- Kepler appointed Imperial Mathematician. Rudolph mainly
interested in astrology, but Kepler gave “advice”.
27
28
Nested Polyhedra from Kepler’s Mysterium Cosmographicum
(1596)
•To achieve this he used the naked-eye sighting-instrument enhanced
observations of Tycho Brahe.
•Most accurate of the time; thought to be good to 2 minutes of arc
(one-eighth the size of moon) – actually 4 arc min.
•When checking some additional observations using his vicarious
hypothesis, he discovered an 8 minute of arc discrepancy [in the
latitude].
•Kepler devised a very good heliocentric model using circular
motions that yielded predictions to 2' accuracy – GENERALLY.
•He found, upon further data-mining of Tycho’s observations, that
his model with circles could be off by 8 minutes of arc in a few cases
– which was outside the “experimental” error.
•Chapter 17 begins with:
Who would have thought it
possible? This hypothesis, which so closely agrees with the observed
oppositions [of Mars], is nevertheless false.
- Kepler believed he had insight into God’s mind and His plan for
Creation
- He used three-dimensional spheres inscribed in the five regular
(Platonic) solids (all sides same):
tetrahedron (4 equilateral triangles)
cube (6 squares)
octahedron (8 equilateral triangles)
dodecahedron (12 pentagons)
icosahedron (20 equilateral triangles)
This explains why there are only 6 planets (Mercury, Venus, Earth,
Mars, Jupiter, Saturn).
Sphere thickness accounts for eccentricity, since the spacing was
“not quite” right.
29
30
In Chap 17 of Astronomia Nova, Kepler informed us his predictions
for Mars were in error by 8 minutes of arc.
Did Kepler give up his Neopythagoreanism and mathematical
mysticism with this hard exactitude? NO !
Looked for harmony in sizes of orbits, speeds of planets, but
concluded that the numbers revealed that God did not put
harmony there (and Kepler would not force it).
Finally rejected uniform circular motion for elliptical motion with
changing speeds. Two thousand years of astronomical tradition for
minuscule discrepancy. Kepler went against Plato, Aristotle,
Thomas Aquinas, and Copernicus. But replaced it with an extremely
accurate set of three mathematical laws.
In his Harmonice Mundi (1619) he announced his 3rd law:
(Orbital Period)**2 proportional to (Semi-major Axis)**3.
31
32
Galileo was contemporary of Kepler. Most of you know of his
troubles with the Church. But they are not so relevant to our story as
his "objective" attitude toward math in nature
- After great perseverance, he found harmonious ratios in the
angular velocities of the planets at their closest and farthest
distances from the sun, as viewed from the sun. This revealed
the divine music in the Mind of God.
- Scales of the planetary music. Truly heavenly music.
33
34
Descartes wanted to rid the mind of inherited ideas and start from
first principles to produce a complete philosophical and physical
system of thought. His approach was much more rigorous than its
Aristotelian predecessors. It was pattern much like a mathematical
treatise in the form of its logic.
Produced a great development and synthesis of mathematical
exposition of physical law:
Mathematical Principles of Natural Philosophy (1687)
-Descartes felt that the mind becomes too cluttered by the confusing
spectrum of sensation experiences and authoritative teaching. In his
Discourse on Method (1637), he urged the reader to wash his mind
clear of this confusion and start over with principles that “we clearly
and distinctly conceive as true.”
-This resulted in “Cogito, ergo sum” (I think, therefore I am; i.e., I
exist)
- From God's unchanging immutability, Descartes deduced various
conservation laws for physical motion.
- Gave a correct version for the law of inertia.
35
36
Very, very accurate predictions based on an underlying
understanding. Insightful and fruitful.
God as Urizen, the rational geometrical architect: Neoplatonism
and neopythagoreanism
Derived and deduced much physical behavior, including Kepler's
planetary laws of motion.
Pre-fallen Man (Godlike emanations) had balance of four
qualities/powers
Urizen – Reason
Imagination
Tharmas – Sensation
Los (sol backwards) –
Luvah - Emotion
Blake was VERY opposed to hegemony/dominance of Reason.
Dominance of Reason takes away mystery, awe, wonder. Takes
man away from God.
37
38
•Blake's perceptive reaction to mathematization of the Cosmos. He respected
Newton, but not dominance of reason. Newton's heroic body bends over,
constrained, tunnel vision.
Laplace was the younger associate of Lagrange.
•Newton became concerned with the reception of his ideas, like intellectual
Pandora's box. He believed the solar system would become unstable and be
destroyed by mutual gravitational perturbations until it "wants a reformation".
He showed that the mutual interactions of the bodies in the solar
system produce only cyclic perturbations in their orbits. The
irregularities would not continue to increase as Newton had
thought, but instead would oscillate about their average motions.
•This required God's intervention, since the Universal Ruler "is more able by
his will to move bodies within his boundless uniform sensorium, and thereby
to form and reform the parts of the Universe, than we are by our wills to move
the parts of our bodies". (Opticks, 2nd edition, 1717, Query 31)
•Within a century, Newtonian science, which was to reveal spiritual glory,
changed into a serpent which pursued theological and humanistic philosophy.
Using the mechanistic principles of Newton, “natural philosophers” (a.k.a.
scientists) were eventually able to explain previously inexplicable phenomena
without evoking any spiritual agents.
•This concordance with phenomena was so appealing that the scientific
methods codified by Newton became the new paradigm.
39
40
What of neo-Pythagoreanism today?
Dirac related a story of Erwin Schrodinger, developer of
quantum wave mechanics.
Schroedinger developed a fully relativistic equation to describe
the electron in hydrogen, but his predictions did not quite agree
with experiment because spin effects could not yet be observed.
In discouragement, Schrodinger published a non-relativistic
version. Experiment later showed that his original equation was
right, but Klein and Gordon had already published the relativistic
wave equations.
41
42
Do you hear Kepler with his polyhedra?
What’s missing ?
43
44
Missing particle is baryon (B=1)
with S = -3 and T3 = 0
Electrical charge is given by
Q= T3 + (S+B)/2
Q = 0 + (-3+1)/2
= -1
Ω¯ !
45
46
The idea of a Mathematical God has received serious
consideration by some of the great minds of the ages:
Plato, Aristotle, Aquinas, Kepler, Galileo, Einstein, Heisenberg –
noble company, indeed
47
48
49
50