The Mind of God Session Notes

based on The Mind of God; Paul Davies; ©1992, Orion Productions.

  1. Discussion Issues

    1. What is God's role in creation?
    2. If science can explain the beginning of the Cosmos and the Laws of Nature without resort to God, is God necessary?
    3. Where is God in the creation? In the gaps? Or ...

  2. Can the Universe Create Itself?

  3. What Are the Laws of Nature?

    1. The Origin of Law
      • the polytheistic ancients viewed the world as a living organism of contending spirits and gods, rather than as a machine with governing laws.
      • Some definitions
        teleology
        idea of a physical system seeking out or being directed towards a final goal
        Material Cause
        The objects that make a cause possible: the bricks of a house
        Formal Cause
        The form or shape of the cause: the shape of the house
        Efficient Cause
        means whereby the cause happens and becomes its form: the builder
        Final Cause
        the purpose of the cause
      • The monotheistic religions, though, separated God from Creation, and creation was subject to laws imposed by God.
      • Laws are imposed on nature, not inherent in nature (as in polytheism).
      • Therefore, studying the laws of nature is uncovering God's rational design (so believed Tycho Brahe and Johannes Kepler; Descartes and Newton)
      • God the Cosmic Mathematician and Engineer
        • Does God wind the universe up and sit back and watch it run?
        • Does God supervise it day by day?
          Newton believed the universe was saved from gravitational disintegration only by a perpetual miracle of God: God of the Gaps!
      • Descartes and Leibniz (contra-Newton): God is the fountainhead and guaranator of the total rationality that pervades the cosmos.
      • Modern scientist: unreflective on origins of the laws
      • Yet, Western theology laid the groundwork for the scientific enterprise
        1. Divine being legislating law
        2. Creation can be mined for God's mind
        3. Reductionism works
      • Now we are learning that the holism of the East is important to understand nature
    2. The Cosmic Code
      • The rise of science and the Age of Reason: a hidden order in nature, mathematical, that could be discovered.
      • data -> theory (decoder) -> laws
      • crack the Cosmic Code -- an intuitive process
      • universe has a hidden, mathematical order that explains phenomona
      • senses -> World-Soul -> Platonic Forms (Plato)
      • laws => cosmos (not chaos)

      Why might the laws be so hidden?

    3. The Status of the Laws Today
      • Did God "encode" the laws for us to discover?
      • Nothing (so far!) in science indicates that there was an independent encoder
      • So, where did the laws come from? Are they simply there?
      • What is a "law"?
        1. description of a regularity of nature
        2. deduced
      • Have we just imposed the regularities as we impose pictures (constellations) on the stars?
        1. No, regularities are an objective mathematical fact.
        2. And they help us uncover new things about the universe:
          • Newton's Law of Gravity explains more than planetary motion. It also explains ocean tides, motion of spacecraft, etc.
          • Laws of Nature are deep connections among physical processes
      • Properities of the Laws (previously properties of God):
        1. Universal: they work everywhere at everytime
        2. Absolute: independent of the observer and state of the universe; these are dependent on the laws
        3. Eternal: defined with platonic forms
        4. Omnipotent: all-powerful wrt physical systems
          Omniscient: always know the state of a physical system
      • The status of the Laws:
        • Discoveries or inventions?
        • Are they intrinisic to Nature, or just mathematical models to which an alien could invent alternative?
        • Do the laws transcend the physical world? That is, can they be observed separate from the physical processes they affect?
        • software vs hardware :: laws vs physical states
        • is there a "cosmic software system" independent of the "hardware"?

      What is the difference between the Laws of Nature and God?

    4. What Does It Mean for Something to "Exist"?
      • An object we sense has an existence independent of our senses (it just doesn't appear when we turn our attention to it)
      • Therefore, objects that we do not sense (such as atoms) can also exist: we detect them indirectly
      • Further, there exists energy fields -- even more nebulous
      • Similarly, concepts such as citizenship and information and software exist, though they are not physically detectable; nonetheless they are highly influential
      • And subjective phenomena: dreams and the imagination, emotions, memories, sensations
      • The soul; religion
      • Cultural: music, literature, etc
      • The laws of physics can therefore have an independent existence
      • The "laws" we know are only approximations of the platonic laws; we continue to converge on them.
      • But are they really independent and not just pecular to our culture and place in the universe?

      What, if anything, exists "behind" the Laws of Nature?

    5. In the Beginning
      • Initial conditions depend on the environment that started a process
      • The cosmic initial conditions (@Big Bang), however, have no environment with which to begin: they are "given" like the laws
      • Scientists want to not explain by resorting to special conditions, such as God
      • Therefore, scientists desire to minimize the effect of the cosmic initial conditions (e.g., the cosmos would be similiar within a range of conditions), but the initial conditions are important
      • But why these particular "cosmic initial conditions"? Is there some deep reason for them?
      • Not just any initial conditions will do. So why these?
      • Law of Initial Conditions being sought: how the universe came into existence
      • Laws are platonic forms existing independent of space-time? yes, if there is a law of initial conditions

      Where is God and what is God's role in the universe?

  4. Is Mathematics Real?

    [Mathematics] is also, astonishingly, the language of nature itself. No one who is closed off from mathematics can ever grasp the full significance of the natural order that is woven so deeply into the fabric of physical reality [Davies, p93].
    1. Numbers are Sacred and Mystical
      • Pythagoras: 6th Century BCE; Pythagoreans
        • "Number is the measure of all things."
        • cosmic order based upon numerical relationships
        • certain numbers are mystical
        • "perfect" numbers: sum of their divisors: 6 = 1+2+3
        • divine tetraktus: 10 = 1+2+3+4 (first 4 whole numbers)
        • 4: justice and reciprocity; "a square deal"
        • discovered relationships of length of musical strings to tones: ratios; hence the music of the spheres in pythagorean cosmology
        • numerology: physical world as a manifestation of mathematical relationships.
      • Kepler: God is a geometer; numbers have mystical significance
      • modern physics: cosmos rationally ordered according to mathematical principles.
      • Note biblical numerology:
        Number Meaning
        one unity of God
        two two natures of Christ;
        two thieves crucified with Jesus
        three trinity; completeness;
        three parts of Creation: heaven, earth, hell;
        third hour when Jesus was crucified (MK15:25);
        Jesus rose on the third day
        four four cardinal directions;
        four corners of the earth;
        four winds;
        four rivers of Eden;
        four gospels and four Evangelists
        40 (4x10) a long period of time;
        40 years = a generation
        five the five loaves and the five thousand (a multitude);
        five months in Revelations = a limited time
        six incompleteness;
        sixth hour when darkness descended at the crucifixion (MK15:33)
        666 incompleteness or imperfection three times = anti-trinity
        seven completeness and perfection;
        seven churches of Revelations;
        seven days of Creation;
        seven deacons;
        seven gifts of the Holy Spirit
        21(3x7) absolute perfection
        49 (7x7) jubilee
        70 (7x10) the time to completeness
        (777) divine perfection
        (perfection three times juxapositioned = trinity)
        eight eighth day of Creation
        nine the nine ungrateful lepers (LK17:17);
        the ninth hour when Jesus died (MK15:33)
        ten the ten lost tribes of Israel
        eleven the company of loyal apostles
        twelve tribes of Israel (all of the society);
        twelve apostles;
        the number of perfection (3 x 4)
        144,000(12x12x1000) 12 tribes x 12 apostles x 1000 (infinity) or
        12 tribes squared (perfection) x infinity = the whole people of God
      • 7 is lucky; 13 is unlucky
      • circle is eternity
      • Newton and others discovered the significance of time also: number, time, space (geometry); motivated by navigation; resulted in the Calculus
      • we intuit there is a cosmic significance to numbers and shapes
    2. Math Is Open Ended
      • As Kepler and the Greeks saaw God as Geometer and Newton saw God as a Watch Maker, we see God as a Computational Process
        • Universe: music of spheres : cosmic clock : cosmic computer
        • mixes up God and nature: paganism
          1. All these metaphors risk idolatry
          2. God is utterly other
          3. metaphors are necessarily limited
        • Laws of Nature: Computer Program
        • Unfolding Events: Output
        • Initial Conditions of the Universe: Input
      • Universal Computer: execute any computable mathematical function
        • computable: solvable by a finite program (though may require infinite steps; e.g., irrational numbers)
        • formalism
          1. mathematical rules applied to symbols
          2. formal manipulation of symbols
          3. no necessary relationship to physical worls
          4. e.g. rules of arithmetic
          5. abstract without physical meaning
          6. hoped to apply to proving theorems
        • formalism undermined in 1931 by Gödel
          1. there exist math statements for which there is no proof (i.e., not solvable by formalism)
          2. there exist undecidable propositions
          3. therefore, math is open-ended
      • Gödel: undecidable theorems spring from paradoxes
        • This statement is false
        • Self-referential statements
      • John Barrow (p. 101): If religion is defined to be a system of thought which requires belief in unprovable truths, then mathematics is the only religion that can prove it is a religion.
      • Uncomputable: There does not exist a finite number of steps to compute
      • Universal Turing Machine : logical model of any computer
        • inpute specifications of any Turing machine
        • compute function
      • "Halting Problem": to be able to tell ahead of time if an algorithm will stop or loop infinitely on a certain input.
        • Cannot be done
        • therefore cannot systematically determine decidable vs undecidable
    3. The Unreasonableness Reasonableness of Mathematics
      • irrational numbers
      • imaginary numbers
    4. Platonic Connections -- see Fibonacci Gold below
    5. Emptiness
      1. In mathematical set theory, a set is a collection of things taken as a whole.
      2. The simplest, most fundamental set is the set containing nothing, called the empty set.
      3. It is the whole that contains nothing.
      4. A subset of a given set is a set that contains some or all of the elements in a set.
      5. One subset of any set is the set of no members, {}, the empty set.
      6. The empty set is in or a part of everything, even of itself (since a set is a subset of itself).
      7. It is the only subset that is a subset of everything. The empty set is, therefore, ubiquitous.
      8. It is also unique; there is only one empty set.
        • Two sets are identical if they have exactly the same elements.
        • Now if two sets have no elements, they have exactly the same elements.
        • Thus, they are the same and one occurrence of the empty set is the same set as any other occurrence of the empty set.
      9. Since the empty set has no elements, adding its elements (which there are none) to any other set just brings us back to the set.
        • Uniting emptiness to anything results again in that anything.
        • Emptiness is already part of everything and, being unique, adding it again yields nothing new.
        • Emptiness plus you = you.
        • It is an essential part of us.
      10. Similarly since the empty set has no elements, it has no element in common with any set.
      11. Therefore, the empty set is utterly other. It is dissimilar to any set.
      12. Thus, everything has nothing in common with nothingness: though emptiness is within everything, we have nothing in common with it; it is utterly other.
      13. Two sets are utterly different if they have no elements in common; they are disjoint.
      14. The intersection of two disjoint sets is the empty set.
        • Since the empty set is a subset of any set and the empty set is utterly other, two utterly different things have the utterly other in common.
    6. Plethora
      • Mathematics of Infinity: Georg Cantor
      • A circle is a panagon: a polygon with infinite sides.
      • Number of reals > number of rationals:
        #(reals)=2#(rationals)
      • The Fabric of Infinity:

Mathematics is mystical!

Fibonacci Gold

Chambered Nautilius Shell,
courtesy of the Georgia Shell Club

What do the chambered nautilius shell, the florets of sunflowers and the quanta energy level of an atom have in common? We are looking for an expression of God's mystical presence. Our most immediate and naive entry to God's presence is in ordinary human sensual exprerience. There we can see God in a creation's beauty and utility. We encounter God's love in the beautiful miracle of a shell or a flower. The care and complexity of evolution could only be designed by a graceful and loving God. Yet beyond our quinsensual experience is a whole mystical world of Divine presence. In our technofunctional world, often bereft of the transcendent, the mathematician can be a thaumaturge who mediates the mystical in an orderly and efficacious way.

To find these deeper, mystical expressions of God, we must enter the MathSpace. The MathSpace is faërie, a place of enchantment and excitement where we can explore strange new lands and denizens. We encounter it with our five senses only obliquely at the shimmering sunrise or the haunting sunset. Thought, we can explore the MathSpace intellectually, we feel and smell and see the MathSpace with our spiritual senses and mathematical imagination.

We can pass into the MathSpace by abstracting the mathematical nature of a creation such as the Nautilus shell. In the MathSpace, we will discover God's design, order, and creativity. The nautilius shell is a logarithmic spiral, expressed as r = eat. Now, a logarithmic spiral is intimately connected to a golden rectangle [Huntley, Pickover], which is based on the golden section. The golden section is a division of a line segment AB into two subsegments AC and CB, such that AB/AC = AC/CB = Phi = 1.61803... Now a golden rectangle is found as follows:

Draw a square ABCD. From the midpoint of AB draw a line to vertex C. Now rotate this line segment until it is conincident with AB. It forms the long side of a rectangle whose short side is coincident with the square. The length of the long side equals the length of the square's side times Phi. This is a golden rectangle:

Goldern Rectangle and Logarithmic Spirial

Now a logarithmic spiral can be inscribed in the golden rectangle as follows: The original square in our construction overlaps the rectangle on one side. The remaining rectangular section is also a golden rectangle. On its short side construct a square. Repeat with each smaller remaining golden rectangle ad infinituum. Now inscribe a quarter circle corner to corner in each square. This will construct a logarithmic spiral. God has placed this beautiful spiral, not only in the chambered nautilius shell, but also in many other creations. See Fibonacci Numbers and Nature.

Thus, Phi, the golden ratio, is an attribute of a logarithmic spiral and by extension of the chambered nautilius shell. In the MathSpace we discover God's underlying connective tissue. The florets of a sunflower form two opposing logarithmic spirals. Unlike in the sensual world of biology and physics, in the MathSpace a chambered nautilius shell, a sunflower, and the coclea of the ear (which is also a logarithmic spiral) are related by being described by logarithmic spirals. And, since a logarithmic spiral embeds a Phi, the shell, ear, and flower inherit the attributes of the golden ratio.

Further, related to Phi are the Fibonacci Numbers. The series of Fibonacci Numbers is calculated as

F0 = 0
F1 = 1
Fn = Fn-1 + Fn-2

This generates 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Fibonacci numbers occur frequently in nature. See Fibonacci Numbers and Nature. For example, each successive generation of the family tree of bees is a Fibonacci number. In a colony of bees:

Thus, female bees have two parents, a male and female, but male bees have only one parent, a female. The family tree of a male bee forms a Fibonacci sequence:

  1. He has 1 parent, a female.
  2. He has 2 grandparents, since mom had a father and a mother.
  3. He has three great grandparents, since grandma had 2 and grandpa had one parent.
  4. He has 5 great great grandparents.
  5. ...

Similarly a female bee's family tress is a Fibonacci sequence:

Gen. # Gender
1
F
2
F
M
3
F
M
F
5
F
M
F
F
M
8
F
M
F
F
M
F
M
F
13
F
M
F
F
M
F
M
F
F
M
F
F
M

Now how are Fibonacci numbers related to Phi? If you take successive ratios. fn+1/fn as n increases to infinity, the ratio approaches Phi:

Fibonacci Gold Theorem,

Thus a logarithmic spiral, being inscribable in a golden rectangle defined by Phi has an infinite Fibonacci sequence embedded in it. As we can see the ordinary sensual experience and the MathSpace intertwine. Examining shells or flowers reveals a logarithmic spiral. Studying a logarithmic spiral reveals Phi. Exploring in the MathSpace reveals Phi related to Fibonacci Numbers. Having discovered Fibonacci numbers, we discover that they describe many natural, sensual creations, such as bee families. The MathSpace leads us to more connections among God's creations. These connections, of Fibonacci numbers and the Golden Mean hint at a cosmic interconnection and oneness in creation.

References

  1. Matthew Fox; The Coming of the Cosmic Christ; Harper & Row, Publishers; San Francisco ©1988.
  2. H. E. Huntley; The Divine Proportion: A Study In mathematical Beauty; Dover Publications, Inc. ©1970.
  3. Clifford A. Pickover; The Loom of God: Mathematical Tapestries At The Edge of Time; Plenum Press, New York ©1997.
  4. Tamburello; Ordinary Mysticism
Rev. John A. Mills, Pastor, First Congregational Church, Closter, NJ fcclostr@cwn.com