Chapter 5: Eternity, time and Hilbert space

Synopsis

Physical states are represented in Hilbert space by rays, mathematical objects similar to musical notes defined by their rate of change of phase or frequency. The ideal mathematical representative of a ray is the polar form of a complex number. We establish basic set of distinct rays in Hilbert space using fixed point theory. Since this space is linear we make more complex vectors by adding basis vectors together, the process called superposition. Mathematics is very useful for describing physical states, but some mathematical ideas like infinity cannot be realized physically.

5.1: Is time primordial?

5.2: Complex numbers

5.3: Fixed point theory

5.4: Logical constraint

5.5 The limits to mathematics

5.1: Is time primordial?

Aquinas argues that God is immutable and therefore eternal. Like all his derivations of the properties of God this one is built around the idea that God is pure action, devoid of potential. Since Aristotle taught that motion is the transition from potential to actuality God, without potential, cannot move. Aristotle differed from Parmenides and Plato in that he took motion seriously and developed his theory of matter and form (hylomorphism) to explain it. He developed the theory of the unmoved mover to drive the hylomorphic transition of forms. His argument that the prime mover is a divine being of pure action was used by Aquinas to model the Christian God. Aristotle: Metaphysics book XII: The life of God (1072b14 sqq.), Hylomorphism - Wikipedia

Motion is intimately connected to time. Aristotle defined time as a number of motion relative to before and after. (Physics, 219b1–2). The measurement of the frequency of the electromagnetic radiation emitted and absorbed by matter was the first empirical step toward quantum mechanics. Energy, in quantum theory, is the product of the quantum of action, h and frequency: E = hf. The first step in the creation of the Universe within the eternal initial singularity appears to be the emergence of something like time. How are we to imagine this?

The only resource available in the initial singularity is omnipotence, like pure action the ability to do anything not self-contradictory. Since it has no structure to embody control its actions must be random, like the sporadic ticks of a clock based on radioactive decay.

It has been held since ancient times that mathematics is the key to understanding the world. The discovery of the numerical relationship between musical tones and the lengths and tensions of the strings on stringed instruments has been attributed to the Greek Pythagoreans. Music as largely a matter of frequencies, events per unit of time and intervals, the inverse of frequency, the time between events.

From a quantum mechanical point music explains everything. The primary role of the quantum mechanism is to find stationary points, like musical notes, in Hilbert space, the space of perpetual motion driven by omnipotent naked gravitation. Mathematically the perfect symbols to describe repeated motion are complex numbers.

5.2: Complex numbers

Time, frequency, interval and stillness are critical measurable quantities in both music and physics. The complex numbers, which are inherently periodic, are naturally well suited to representing cyclic events like musical notes and vibrations. Complex numbers are written z = a + bi, wherei is the complex or imaginary unit equal to the square root of –1, √-1. They were discovered in the sixteenth century. Although they cannot be used to count real things like sheep they can be a very handy intermediate step in a calculations and are perfect for talking about waves. Complex number - Wikipedia

A coarse measure of frequency is the Hertz (cycle per second) named for Heinrich Hertz who showed that Maxwells equations represent real electromagnetic radiation like light and radio waves.

A finer measure of waves is phase, often represented by the Greek letter phi, φ which corresponds to the English f. The phase of a wave is measured in degrees from 0 to 360º or radians from 0 to 2π.

Complex numbers may also be represented using a circle on the complex plane. This is their polar form z =r&heta;. It comprises a radius vector, a real number r, the radius of the circle (the modulus), and an angle, φ, (the argument or phase) running from 0 to 2π. We write z = rφ. This representation emphasizes the periodic feature of complex numbers.

The remarkable thing about complex numbers is that one revolution may represent one complete quantum of action. In spacetime the quantum of action has dimensions of angular momentum. Louis de Broglie saw that the superposition of waves produces fixed points ie standing waves. The atomic orbits proposed by Niels Bohr are standing waves. Together with the quantum of action, these standing waves define frequencies so precisely that they can be used to design atomic clocks that keep time to something like a second in the age of the Universe.

In music instruments are usually tuned to the same set of fundamental frequencies, the scale. The different sounds made by different instruments are due to the superposition of overtones, frequencies which are multiples of the fundamental. Integral multiples produce standing waves, but much of the richness of sound arises from fractional overtones.

5.3: Fixed point theory

Georg Cantor defined a set as any collection into a whole of definite and separate objects of our perception or our thought. The members of a set are called its elements, The set idea is very powerful and has served as a foundation for arithmetic and mathematics in general. Cantor’s definition of a set given here is a bit vague, and led to some trouble. Now sets and their properties are defined axiomatically. Since we are doing theology rather than mathematics Cantor’s definition will do. Set theory - Wikipedia

Operations on sets, called functions or mappings, proceed by establishing one-to-one correspondences between elements of the sets. Since the elements are definite and separate, that is unique, these correspondences can also be unique and well defined. We can map different sets to one another and discover if one has more elements than another by finding elements without a pair. We can simulate motion by mapping a set onto itself, mapping each element onto some other element. Function (mathematics) - Wikipedia

Under certain conditions, this mapping leads to an interesting result in the area of mathematics known as topology. We may find that there is one point that cannot be moved, it must be mapped onto itself. It is a fixed point. There are three conditions that guarantee this situation.

First the set must be continuous: its elements are close to one another as possible, no gaps. Second it must be convex: no empty zones; and third it must be compact, it contains its own boundary. In this case, the Brouwer fixed point theorem applies. If we call the mapping function f (x) and the elements x, then there must be an element x such that f(x) = x. Alan Turing, who invented the computer, showed that there are a countable infinity of computable functions. In Chapter 11: The axioms of abstract Hilbert space we identify these fixed points in the initial singularity with the basis vectors of a Hilbert space. Brouwer fixed point theorem - Wikipedia, Turing machine - Wikipedia

5.4 Logical constraint

We are trying to build a new theory of creation to replace the old idea that Yahweh already had a design in mind when they said let there be, fiat. In Chapter 1 I have explained that given our modern ideas about the representation of information the absolutely simple divinity imagined by Aquinas cannot be omniscient or even have any knowledge at all. Our idea of god, the initial singularity, is omnipotent but since they have no ideas to guide them their actions must be random. As Aquinas points out, the only constraint on an omnipotent agent is that it cannot do anything impossible. In logical terms, it cannot actually make some situation pidentical to its own contradiction, not-p.

Here we need to make a distinction between living and dead, or, we might say, between dynamic and kinematic. This is confusing because the term kinetic energy (in Latin vis viva) sounds very much like kinematic. Something dynamic, like a live wire, will act if it is not impeded. Physicists say it has electrical potential. If you touch a live wire you will get a shock. If you turn on the switch, the light will come on. You have connected the light to the electrical potential. If you are stuck in your car with a snake it has the potential to bite you so you must avoid it.

A kinematic event is something passive, like a puppet, driven by agent outside itself pulling the strings. The atoms in the puppet might be alive, but as a whole it cannot act by itself. A dynamic event, on the other hand, is something performed by a living entity. As Aristotle defined it, life is self motion.

Mathematical theorems establish a logical connection between a set of assumptions and a conclusion. In the case of the Brouwer theorem, we find that if a certain set is mapped onto itself, we will find one point that cannot be moved. Since a set is a formal entity, it cannot map itself onto itself but if, as we maintain here, the initial singularity is dynamic, a living god, not a puppet or graven image, we may understand that it is capable of mapping itself onto itself, that is becoming conscious. Gravitation is non-linear because it acts on itself.

This is consistent wth the theory of the Trinity first developed by Augustine. The Son of God is God the Father’s conscious image of themself. On the assumption that God is pure act everything is dynamic in God, and we may assume that the Father’s image of himself is also really a person, the Son.

5.5: The limits to mathematics

Cantor’s invention of transfinite numbers caused a stir in the intellectual world. Theologians objected that only God is infinite. Hilbert, following Cantor invented formalism to deal with this problem. The rules are simple. Mathematicians can freely invent axioms and work out their consequences as long as they maintain logical consistency. In mathematics logical consistency is equivalent to existence. Hilbert thought that this would mean that there could be no ignorance in mathematics, non ignorabimus, we will not be ignorant. Transfinite numbers - Wikipedia, Formalism (mathematics) - Wikipedia

His dreams were shattered by Kurt Gödel and Alan Turing. Hilbert thought consistent mathematics would be both complete and computable. Gödel showed that there are unprovable statements, it is not complete. Turing showed that there are incomputable functions. It is not computable

Cantor began his construction of transfinite numbers with the set of natural numbers N = {0, 1, 2, . . .} where { . . . } is the conventional notation for a set. This set is infinite because there is no last natural number. It is called countably infinite. Cantor coined a special term for the cardinal of N using the first letter of the Hebrew alphabet, ℵ₀. This he called the first transfinite number.

Turings proof showed that there are incomputable functions, but as the computer industry has shown, there are still plenty of functions to work with. The actual fixed point x which we obtain on applying the fixed point theorem will depend on the function f. Since there is a countable infinity of turing computable functions, f (x), we might expect that there are also ℵ₀ fixed points in the initial singularity. Let us assume that these points constitute a set, a space called Hilbert space, the home of quantum mechanics. This space exists in the initial singularity which also contains time, so we may see Hilbert space as a countably infinite collection of frequencies or sounds.

When the Pythagorean theorem was invented it showed that the diagonal of a unit square was the square root of two √2. It was then proved that √2 cannot be a fraction, a ratio or rational number, a / b. This means that there must be numbers between the fractions, real numbers. It turns out that there are just as many fractions as natural numbers, so Cantor thought that the second transfinite number, ℵ₁ might be the cardinal of the continuum, the number of points it takes to make a continuous line. He then went on to suggest that there would be ℵ₂ points between these points and so on without end, creating an infinite hierarchy of transfinite numbers

Later, when set theory was axiomatized Paul Cohen showed that it could not really tell us anything about transfinite numbers. It turned out that the idea of creating a continuous line out of discrete points did not work. Later we will suggest that the mathematical idea of a continuous geometric line is a mathematical ideal not found in reality. Paul Cohen (1980): Set Theory and the Continuum Hypothesis

An interesting feature of complex numbers is that they form a seamless connection between continuous rotation and the integers, each integer representing a quantum of action. The integers represent fixed points in the sequence of waves, one full turn of a complex number, a quantum of action.

Chapter 5: Eternity, time and Hilbert space

Synopsis

Contents

5.1: Is time primordial?

5.2: Complex numbers

5.3: Fixed point theory

5.4 Logical constraint

5.5: The limits to mathematics

Notes and references

Further reading

Books

Links