vol III Development:

Chapter 3: Cybernetics

page 7: The transfinite oscillator

The transfinite network is a formal structure, eternal and static, like all mathematical objects. The dynamics of mathematics is to be found in the minds of its users. We may think of mathematical objects as fixed points in the mathematical community. Mathematical texts are written down and shared. The consistency of these fixed points is guaranteed by the work of mathematicians seeking to prove or disprove relationships between them.

Here we work on the assumption that the only constraint on mathematics is consistency, and we attribute mathematical existence to any consistent symbolic system. We believe that the same criterion holds for the divine Universe - that any consistent structure is capable of existence if the physical resources are available to represent it. We feel that the transfinite network is a formal system capable of addressing the whole space of consistent symbolism. It also has the maximum possible computing power in the parallel operation of any number of Turing machines. Formalism (mathematics) - Wikipedia

We now turn to a dynamic view of the transfinite network, which we will call the transfinite oscillator. This device embodies the Hindu concept Trimurti, the three forms or modes of the divinity, creator, maintainer, and annihilator. These three features are to be found in the evolutionary process which has brought the Universe from the initial singularity to its current state. Fixed points, are born, exist for some time, and die; creation, maintenance and annihilation. Trimurti - Wikipedia

An oscillator is a system that cycles between the members of a set of fixed points or states. A simple classical pendulum, for instance, cycles between the turning points at each end of its swing. When it is stationary, a pendulum hangs straight down. If some force displaces it from this position, it experiences a restoring force proportional to the displacement. Its oscillations come from the interplay of these two forces: the force of gravity urging it toward the centre of its swing, and the force of momentum carrying it away from its central position. Pendulum - Wikipedia

Quantum mechanics introduced the quantum harmonic oscillator. The pendulum cycles between potential energy (when it is high and slow) and kinetic energy (when it is low and fast). The quantum harmonic oscillator also cycles between kinetic and potential energy. Whereas pendulums swings with a particular fixed frequency (which is why we use them in clocks) the quantum harmonic oscillator has a spectrum of frequencies. Quantum harmonic oscillator - Wikipedia

This spectrum is quantized, and the energy difference between adjacent states is one quantum of energy. One can model the behaviour of a quantum harmonic oscillator using the ladder operators a and a^†. '. . .a^†, in essence, appends a single quantum of energy to the oscillator, while a removes a quantum. For this reason, they are sometimes referred to as "creation" and "annihilation" operators'. Ladder operator - Wikipedia

We define the transfinite oscillator in an analogous way. We imagine a creation operator a^† which carries us from the layer ℵ_n to layer ℵ_n+1 and an annihilation operator a which carries us from ℵ_n to ℵ_n-1.

There are two processes that take us up and down the ladder of complexity. On the one hand we can increase complexity by permutation. On the other hand, we can decrease complexity by considering complex things as sets, that is unities which can be named and manipulated (at least theoretically) as whole. One way to describe sets is by the symmetries shared by their members. So we define the set of humans by the expression H = {x|x is human}. Michael Bradley, United Nations

The power of set theory is that it is indifferent to complexity. The size of sets is governed only by consistency. The set of all sets can have no mathematical existence because it leads to contradiction. Sets are the tool that enabled Cantor to 'tame' the transfinite numbers so that they can be handled. By using one to one correspondence to compare sets, Cantor devised a process to measure their relative sizes. Russell's paradox - Wikipedia

The transfinite oscillator operates between different levels of the transfinite hierarchy. Cantor noted that the process of generating the transfinite numbers is the same at every level of complexity, a 'unitary law'. This I call symmetry with respect to complexity. Arithmetic is symmetric with respect to complexity, and this is also true of quantum mechanics and other mathematical theories based on arithmetic. Cantor: Contributions to the Founding of the Theory of trasfinite Numbers, page 109

The fixed foundation of the transfinite numbers is the set of natural numbers, cardinal ℵ₀. Sets whose cardinal is ℵ_o are said to be countably infinite. ℵ₀ is also the cardinal of the set computable functions. On the other hand, there are ℵ₁ mappings from the natural numbers to themselves, so the overwhelming number of these mappings are not computable. They can only be represented, if at all, by lookup tables.

The existence of symmetries and the formation of sets, however, opens a route to computation with set of higher the higher cardinals ℵ_{n > 0}. This analogous to the techniques used by a teacher in a traditional classroom to deal with the diference of variety between the single teacher and multiple children. By discipline, the variety of the children is reduced to the variety of the teacher.

Nevertheless, every action in the transfinite network is ultimately unique, identified by the unique states before and after the action. All symmetries within the system are broken. Where there is not a computable path defining an action there is uncertainty. No matter how hard we try to control things, there is always a small probability of error. Feedback may be used to identify and correct errors, but feedback requires additional processing and this itself is subject to error.

A consequence of this is that time becomes of the essence. The rate of processing, that is the rate of action, is measured by energy. We may view every computation as a communication channel. Every computation in a network occurs within the context of other computations which may be seen (from the point of view of a particular computation) as noise, insofar as they may change the fixed points associated with the computation of interest. In communication theory we see this as one symbol being replaced by another.

Communication theory shows us how to transmit information error free in the presence of noise, but the necessary process introduces delay, for two reasons. The first is because the system encoding a message must wait for the source to emit enough symbols to form a packet. Similarly, the receiver must receive a whole packet before it can begin to decode. The second is that encoding and decoding require computation and which also take many operations and therefore time.

The general effect of these constraints is that certainty is associated with slowness and uncertainty with speed. But a system operating in the context of other systems often has time constraints. Tennis players, attempting to return shots from their opponents, have very limited time to control the ball effectively. No matter how good the players are, they inevitably make errors which bring interesting uncertainty into the game.

Quantum mechanics has taught us that all observable processes in the Universe are associated with a certain quantum of action measured in units of Planck's constant, h. No action smaller than h is observed. Quantum - Wikipedia

Quantum mechanics also introduced uncertainty into physics. Before quantum mechanics, many people though that the Universe is a deterministic automaton, an opinion embodied in Laplace's Demon. Laplace's demon - Wikipedia

This uncertainty is often represented by expressions like ΔE.Δt ≈ ℏ and Δp.Δx ≈ ℏ. These equations limit our precision of measurement and prediction. To measure energy more precisely we must observe for a longer time; to measure momentum more precisely, we must observe for a longer distance. Certainty is obtained by extended observation. Uncertainty principle - Wikipedia

Complexification

The ideal classical simple harmonic oscillator such as a pendulum or a mass on a spring operates deterministically at a fixed frequency forever. It exchanges no energy with its environment. Ideally we cannot even watch it lest the momentum of photons change its state.

The quantum harmonic oscillator steps up and down a ladder of fixed equally spaced states. Which step it will actually take is uncertain, but the stepping has a statistical structure which is related to the potential in which the oscillator is operating. The process does not change the states themselves, only their rate of occupancy.

Formally, the transfinite oscillator is free to explore the whole space of consistent states, so that its phase space is isomorphic to mathematics. Not only can it change the occupancy rates of various states, but it can introduce new states. In quantum mechanics, new state spaces are introduced by taking the tensor products of the interacting Hilbert spaces, creating more complex state vectors, that is ordered sets of 'moving parts'. Phase space - Wikipedia

Although the transfinite oscillator moves up and down the ladder of complexity, the principle of requisite variety tends to bias it toward increasing complexity. The principle implies that complex systems can control simple ones but not vice versa. Complex systems are able to make themselves more durable by better control of their hardware. We can see this effect in health care, where better care for our bodies enables us to live longer and more active lives and usually improves our mental state also. Variety (cybernetics) - Wikipedia

The transfinite oscillator shines some light on the relationship between matter and spirit. We take them to be relative terms, defined in the first instance by the lower (material) and upper (spiritual) associated with a location in the transfinite oscillator. Since the spiritual element is more complex, we may say on cybernetic grounds that spirit can control matter. Matter - Wikipedia, Spirit - Wikipedia

If either the spiritual or the material element fails, however, the system fails. Death can come either from failure to control, or lack of something to control: one may starve for lack of food, or due to some personal problem.

(updated 12 January 2019)

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Further reading

Books

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Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1895, 1897, 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.'  Amazon   back

Links

Formalism (mathematics) - Wikipedia, Formalism (mathematics) - Wikipedia, the free encyclopedia, 'In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules. For example, Euclidean geometry can be seen as a game whose play consists in moving around certain strings of symbols called axioms according to a set of rules called "rules of inference" to generate new strings. In playing this game one can "prove" that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules.' back

Harmonic Oscillator - Wikipedia, Harmonic Oscillator - Wikipedia, the free encyclopedia, 'In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: F = kx, where k is a positive constant. If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).' back

Ladder operator - Wikipedia, Ladder operator - Wikipedia, the free encyclopedia, 'In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum.' back

Laplace's demon - Wikipedia, Laplace's demon - Wikipedia, the free encyclopedia, 'We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.' A Philosophical Essay on Probabilities, Essai philosophique dur les probabilites introduction to the second edition of Theorie analytique des probabilites based on a lecture given in 1794. back

Matter - Wikipedia, Matter - Wikipedia, the free encyclopedia, 'Before the 20th century, the term matter included ordinary matter composed of atoms and excluded other energy phenomena such as light or sound. This concept of matter may be generalized from atoms to include any objects having mass even when at rest, but this is ill-defined because an object's mass can arise from its (possibly massless) constituents' motion and interaction energies. Thus, matter does not have a universal definition, nor is it a fundamental concept in physics today. Matter is also used loosely as a general term for the substance that makes up all observable physical objects.' back

Michael Bradley, How we will look back on same sex mariage opponents, 'The idea that black people are by nature inferior was the moral foundation on which both the institution of slavery and the outlawing of racial intermarriage were based. Whether the justification was explicitly religious, as the apartheid regime in South Africa made it, or founded in so-called natural law, which is where the Nazis placed their racial theories, belief in racial inequality has always had an ethical core. Warped ethics, we now agree, but ethics nevertheless. Aristotle, who pretty much invented ethics, also said this: "Some men are by nature free, and others slaves, and that for these latter, slavery is both expedient and right."' back

Pendulum - Wikipedia, Pendulum - Wikipedia, the free encyclopedia, 'A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth.' back

Phase space - Wikipedia, Phase space - Wikipedia, the free encyclopedia, 'In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. , , , back

Quantum - Wikipedia, Quantum - Wikipedia, the free encyclopedia, 'In physics, a quantum (plural: quanta) is an indivisible entity of a quantity that has the same units as the Planck constant and is related to both energy and momentum of elementary particles of matter (called fermions) and of photons and other bosons. The word comes from the Latin "quantus," for "how much." Behind this, one finds the fundamental notion that a physical property may be "quantized", referred to as "quantization". This means that the magnitude can take on only certain discrete numerical values, rather than any value, at least within a range.' back

Quantum harmonic oscillator - Wikipedia, Quantum harmonic oscillator - Wikipedia, the free encyclopedia, 'The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.' back

Russell's paradox - Wikipedia, Russell's paradox - Wikipedia, the free encyclopedia, 'According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.' back

Spirit - Wikipedia, Spirit - Wikipedia, the free encyclopedia, 'The English word spirit (from Latin spiritus "breath") has many different meanings and connotations, most of them relating to a non-corporeal substance contrasted with the material body.' back

Transfinite number - Wikipedia, Transfinite number - Wikipedia, the free encyclopedia, 'Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.' back

Trimurti - Wikipedia, Trimurti - Wikipedia, the free encyclopedia, 'The Trimurti (English: ‘three forms’; Sanskrit: त्रिमूर्ति trimūrti) is a concept in Hinduism "in which the cosmic functions of creation, maintenance, and destruction are personified by the forms of Brahmā the creator, Vishnu the maintainer or preserver, and Śhiva the destroyer or transformer," These three deities have been called "the Hindu triad" or the "Great Trinity", often addressed as "Brahma-Vishnu-Maheshwara."' back

Uncertainty principle - Wikipedia, Uncertainty principle - Wikipedia, the free encyclopedia, 'In quantum mechanics, the uncertainty principle, also known as Heisenberg's uncertainty principle, is any of a variety of mathematical inequalities[1] asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position x and momentum p, can be known.' back

United Nations, Official UN Universal Declaration of Human Rights Home Page, 'The Universal Declaration of Human Rights (UDHR) is a milestone document in the history of human rights. Drafted by representatives with different legal and cultural backgrounds from all regions of the world, the Declaration was proclaimed by the United Nations General Assembly in Paris on 10 December 1948 General Assembly resolution 217 A (III) (French) (Spanish) as a common standard of achievements for all peoples and all nations. It sets out, for the first time, fundamental human rights to be universally protected.'' back

Variety (cybernetics) - Wikipedia, Variety (cybernetics) - Wikipedia, the free encyclopedia, 'The term Variety was introduced by W. Ross Ashby to denote the count of the total number of states of a system. The condition for dynamic stability under perturbation (or input) was described by his Law of Requisite Variety. Ashby says: Thus, if the order of occurrence is ignored, the set {c, b, c, a, c, c, a, b, c, b, b, a} which contains twelve elements, contains only three distinct elements- a, b, c. Such a set will be said to have a variety of three elements. He adds The observer and his powers of discrimination may have to be specified if the variety is to be well defined. Variety can be stated as an integer, as above, or as the logarithm to the base 2 of the number i.e. in bits.' back