In sessions 2 through 4, we consider five "contingencies" or "dependencies" which arise in physics, which physics cannot explain beyond accepting them as "brute facts." These unexplained contingencies or dependencies have been suggested as "rumors" of God:

1. The laws of physics.

Even if physics does eventually achieve its grand quest of a "Theory of Everything," (see below), the anticipated elegant and beautiful mathematical theory ("the laws of physics") will itself remain a "brute" fact, its source unexplained, the realm where it "exists" unknown

2. The "boundary" of the universe at time = 0 (the "Big Bang") of classical cosmology. (Caveat: there are hints physics may soon have an explanation for this "boundary.")

3. The existence of all of space-time.

Why is there something and not nothing?

Even if physics does explain the "boundary" at time = 0, the explanation will still just be a theory that cannot explain what "breathed fire" into the equations to make theory a manifest reality, and what "sustains" that reality today

4. The Anthropic Principle. The laws of physics and the "initial conditions" of the universe near time = 0 appear to be incredibly fine-tuned to produce life. Why? Some possibilities:

a Designer

Many Worlds

an "Observer-created" Universe

5. The Ground of Physical Reality of Quantum Physics

the impenetrable "boundary" of physical being encountered in Quantum Physics: a ground of being teeming with latent possibility and potentiality, not yet manifest, not yet real.

In this session, we discuss the first two of these "contingencies."

The universe has been found to obey laws of great mathematical beauty and elegance.

Some quotes:

“The book of nature is written in mathematical language”

- Galileo

“the universe appears to have been designed by a pure mathematician"

- astronomer James Jeans

“. . . It is more important to have beauty in one’s equations than to have them fit experiment. [Since further developments may clear up the discrepancy]”

- Paul Dirac

“the only incomprehensible thing about the universe is that it is comprehensible”

One of the great quests of modern physics has been to find a theory that will unify the four fundamental forces of nature:

gravity

the electromagnetic force

the weak force

the strong force

This quest is based in part on an aesthetic conviction that such an elegant symmetry and ultimate simplicity must exist in nature.

So far:

the Glashow-Weinberg-Salam Electroweak theory in the late 1960's unified the electromagnetic force and weak force.

the general shape or outline of a theory to unify the Electroweak force and the Strong Nuclear force -- a Grand Unified Theory" (GUT) -- seems clear, although the details are still not worked out.

The term "Standard Model" refers to the two present theories which are the foundation of our knowledge of elementary "particles:"

Electroweak Theory

Quantum Chromodynamics, the theory of the Strong Nuclear Force

The best theory of the gravitational force we have today is Einstein's General Theory of Relativity. Most of the classical work in cosmology, including the "Big Bang" and the "initial singularity" at time = 0 that we will discuss today arises from Einstein's General Theory of Relativity

the General Theory of Relativity however does not include the findings of quantum physics, and hence is still not a complete theory of the gravitational force.

a theory incorporating quantum physics effects into Einstein's General Theory still eludes physics, and so far, proposed theories, such as Hawking-Hartle Theory of Quantum Gravity, are still quite speculative

the shape of a theory that possesses the properties that might make it a candidate for a Quantum Theory of Gravity or as a Theory of Everything is actively being pursued. Candidates include:

supersymmetry theories

supersymmetry theories with "local supersymmetry" are candidates for a quantum theory of gravity and are often called "supergravity"

string theory with supersymmetry = supersymmetric string theory = superstring theory

M-theory

Whatever the final theory, physicist are convinced that:

“The Theory of Everything … would be much more than just a catalogue of physical laws. It would constitute a truly unified description of the material universe, weaving an intricate web of interconnections between its component parts, each one essential to the overall consistency of the whole . . . The Theory of Everything would be utterly compelling in structure, symmetry and elegance”

- Coughlan and Dodd, in The Ideas of Particle Physics, 2nd ed, Cambridge University Press

The nature of the "existence" of numbers, geometric figures fascinated the ancients.

Using the example of the geometric form of a triangle, Plato in his "Allegory of the Cave," suggested two realms of reality:

1. sensible world (the dark cave we live in today where we see only shadows)

2. intelligible world of Forms or Essences

He argued that none of us has ever seen a "perfect triangle" of three straight lines with angles that all add up to 180 degrees -- all we have ever seen are imperfect imitations, approximations drawn on a chalkboard. Yet all of us know what a perfect triangle is. How can that be? Where did such knowledge of a perfect triangle come from? He suggested we all have access to a world beyond our own sensible world, a world he called the intelligible world

the Shadows in the Cave are the justice, piety or "chalk-board" triangles that we experience in this life

Forms or Essences are the true and perfect Justice, Piety, and Triangle of the transcendent, eternal "intelligible" world

2.3.3. Mathematics : Platonic? Or A Human Invention?

2.3.3.1. Introduction

Is mathematics:

a purely human invention, a product (or aberration) of the peculiar structure of the human brain?

Platonic? Something which exists in a transcendent realm that mathematicians "discover?"

Example: Question: "True or False: 23 is the smallest prime number greater than 20?" (Answer: true!)

was this question true only after the evolution of the human brain (prime numbers being an invention of the human mind)?

or was this question true before human beings ever evolved?

Most mathematicians sense their work as "Platonic," as the exploration of transcendent landscape of mathematical objects.

"There often does appear to be some profound reality about these mathematical concepts, going quite beyond the deliberations of any particular mathematician. It is as though human thought is, instead, being guided towards some eternal external truth – a truth which has a reality of its own, and which is revealed only partially to any one of us."

- Roger Penrose, Oxford mathematician

The nature of mathematical discovery may offer evidence of mathematics' existence in a "Platonic" realm.

I imagine that whenever the mind perceives a mathematical idea it makes contact with Plato's world of mathematical concepts... When one 'sees' a mathematic truth, one's consciousness breaks through into this world of ideas, and makes direct contact with it..."

- Roger Penrose, Oxford mathematician

The breakthrough is often sudden and dramatic and unexplainable.

Example: S Ramanujan

Indian mathematician born in the late 19^{th} century from a poor family

taught himself mathematics

was able to write down a large number of mathematical theorems without proof – as if he had an extraordinary ability to explore the Platonic mathematical landscape, discover and retrieve the realities preexisting there

his results came to the attention of British mathematician G. H. Hardy who was able to prove some of them with great difficulty

Further suggestions of mathematics' "Platonic" nature includes:

Kurt Gödel's* Incompleteness Theorem 1931

given a set of mathematical axioms, there are propositions that cannot be proven as true or false. There exist undecidable propositions.

Gödel felt true undecidable propositions must already exist in the "Platonic" realm

the discovery of mathematical structures that cannot be fully comprehended by any person or even fully revealed on a computer

example: the Mandelbrot Set

* a recent Science, Vol. 298, page 1899, Dec. 6, 2002 has an article on Gödel

Why is there any order, any rationality at all to the universe? Where does this order, rationality (the laws of Physics) come from? This is a question that physics cannot answer.

One possibility:

“the universe, in its rationale beauty and transparency, looks like a world shot through with signs of mind, and maybe, it's the "capital M" Mind of God we are seeing”

Why are these laws comprehensible to the human mind? And why should such laws appeal to our aesthetic sense of mathematical beauty and elegance?

The human brain presumably formed through evolution in response to environmental pressures (hunting for food, avoiding predators, etc.)

Why should the human mind be capable of discerning, understanding and appreciating the mathematical beauty of the laws of physics?

"If beauty is entirely biologically programmed, selected for its survival value alone, it is all the more surprising to see it re-emerge in the esoteric world of fundamental physics, which has no direct connection with biology. On the other hand, if beauty is more than mere biology at work, if our aesthetic appreciation stems from contact with something firmer and more pervasive, then it is surely a fact of major significance that the fundamental laws of the universe seem to reflect this 'something'"

“. . . there is some deep-seated relationship between the reason within (the rationality of our minds - in this case mathematics) and the reason without (the rational order and structure of the physical world around us). The two fit together like a glove.”

- John Polkinghorne

A Christian may speculate that this deep-seated relationship between the reason within and the reason without may be a reflection of human beings being made in the image and likeness of the source of that rationality, God.

Einstein's Special Theory of Relativity unites the two questions about the relationship of God with time and the relationship of God to space & matter, for it tells us that space and time are a single fabric.

In particular, Special Relativity asserts:

The universe consists of a single fabric of "space-time" consisting of innumerable space-time "events" (a "block" universe of space-time)

The "labeling" of a space-time "event" will depend on how fast the labeler is moving: there is no such thing as absolute or universal time

The unity of space-time in Einstein's Special Theory of Relativity: Stand on the ground and watch a rocket ship go by:

Clocks in a speeding rocketship will appear to run slower (time dilation) than your clocks

Yardsticks in a speeding rocketship which lie along the rocket’s direction of motion will appear short (Lorentz contraction) compared to your yardsticks

Two events on opposite sides of a room in the rocketship that occur "at the same time" to people in the rocketship will not appear to occur to you "at the same time." (relativity of simultaneity)

The theory of relativity should not be interpreted as saying that everything is "relative" or (as some in the humanities seemed to have taken as its message), that everything is "subjective," dependent on the point of view of the observer

While relativity does tell us that quantities we once considered as universal and absolute for all observers are in fact "relative" -- quantities such as the length of an object, time, simultaneity – it replaces these quantities with another universal and absolute quantity: the "timelike interval"

The "timelike interval" is a quantity that consists mostly of the time of an event, but with a little of the spatial dimensions of the event subtracted out

This "timelike interval" is the same for all observers in the universe

The amount of time and the amount of spatial dimension that goes into the computation of the timelike interval may vary from observer to observer, but the result is the same for all observers. The "timelike interval" of an event is universal and absolute

The Special Theory of Relativity also asserts that the speed of light (in a vacuum) is absolute, the same for all Observers no matter how fast or slow they are moving. This is also very counterintuitive:

We all agree that if a baseball pitcher can throw a baseball at 100 miles / hour, then if you put him or her on a truck moving at 50 miles / hour (and neglect the effects of air friction), the speed of the thrown baseball measured from the ground will be 100 miles / hour plus 50 miles / hour = 150 miles / hour.

We would never expect however that if the same baseball pitcher releases a "light ball" at the speed of light from a truck moving at half the speed of light, that the speed of the "light ball" measured from the ground is not 1 times the speed of light plus 0.5 times the speed of light = 1.5 times the speed of light, but -- shockingly -- still just 1 times the speed of light:

Until very recently, most scientists in the modern era have believed the universe was static and eternal.

There were hints of problems:

If the universe was eternal, why hadn't gravity pulled everything to together?

one possible answer -- the universe is infinite, so there is no "central" point for matter to gravitate to.

However, if the universe was infinite and eternal:

wouldn't the forces of gravity add up to be infinite?

wouldn't the surface of the sky be as bright as the surface of a star (Olber's Paradox)

Models of the Universe based on Einstein's General Theory of Relativity (his theory of gravity that superceded Newton's theory) also had the problem of the universe collapsing on itself

Einstein had to introduce a "fudge factor," the cosmological constant to keep the universe static

cosmological constant: an anti-gravity force that kept the universe from collapsing upon itself

In the 20th century, observational evidence began accumulative making it untenable to hold that the universe was unchanging in its form:

1920's: Edwin Hubble presented evidence the universe was not static, but expanding (fabric of space itself is expanding, stretching)

extrapolating back from the present rate of expansion suggests that the fabric of space originated some 15 billion years ago , its matter in the form of a great fireball of unimaginable density, temperature and pressure

1965: the 2.7 degree K background radiation (the redshifted glow of the primeval fireball of the beginning of the universe) detected

The early history of the universe can be described by various "eras" that are based on the type of particle that predominated during that era.

1. Planck Era (the beginning of the universe to 10^{-43} sec from the beginning)

During the Planck era, quantum gravity effects dominated. Since we lack a theory of quantum gravity, we can only speculate about the conditions during this time

2. The Era of the Great Unification (from 10^{-43} sec to about 10^{-35} sec from the beginning of the universe)

During this time the Electroweak Force and the Strong Force are combined in the Grand Unified Force

The universe is dominated by freely roaming elementary particles, some familiar to us today, and some very exotic particles that are unseen today, and all these elementary particles are continually inter-converting between each other.

Grand Unified Theory (GUT) Time

At 10^{-35} sec, the "Grand Unified Symmetry" breaks, and the Grand Unified Force becomes two separate forces, the Electroweak Force and the Strong Force.

What is today the present observable universe measured about 1 mm (1/25th of inch) in diameter at this time

3. The Quark Era (from 10^{-35} sec to 10^{-6} sec from the beginning of the universe)

Quarks, anti-quarks and gluons (the particles that mediate the Strong Force) dominate the universe)

At 10-^{10} sec (The Electroweak Time), the Electroweak Symmetry breaks and the Electromagnetic Force and the Weak Force appear as distinct forces

For the first time, the elementary particles making up matter have the properties we observe in elementary particles today.

4. Hadron Era (from 10^{-6} sec to 10^{-4} sec from the beginning of the universe)

At 10^{-6} sec, the universe has cooled enough that quarks, antiquarks and gluons can no longer roam as free particles. Quarks and antiquarks annihilate each other (signaling the beginning of the Hadronic Era), and combinations of quarks form, called hadrons. There are two types of hadrons:

baryons, composed of 3 quarks or 3 antiquarks. By the end of the hadronic area, the most common baryons are the familiar proton and the neutron.

mesons, composed of 1 quark and 1 antiquark

5. Lepton Era (from 10^{-4} sec to 3 sec from the beginning of the universe)

At 10^{-4} sec, leptons begin to dominate the universe. Leptons include: the familiar electron, the positron (the anti-electron), and neutrinos

6. Photon Era (3 sec to 2 million years from the beginning of the universe)

At 3 sec, photons begin to dominate the universe.

Soon after the beginning of the photon era, at about 200 sec, the synthesis of light elements begins

Near the end of the photon era, about 800,000 years from the beginning of the universe, electrons begin to combine with nuclei and form electrically neutral atoms. Without charged particles to scatter photons, the universe becomes transparent. This is the Recombination time. After the recombination time, there was no further scattered of photons, and what we measure as the 2.7 degree cosmic microwave background is the redshifted glow of the photons that filled the universe before the recombination time.

7. Matter Era (2 million years from the beginning of the universe to the present, about 15 billion years from the beginning)

After radiation and matter had "decoupled" and the universe had became transparent, large scale density fluctuations in matter began to grow that lead to the formation of galaxy clusters and galaxies.