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vol 6: Essays

The unreasonable effectiveness of mathematics revisited

Outline

0. Abstract
1: Introduction: Mathematical theology
2: Fixed points theory and quantum mechanics
3: The fixed point of the Universe
4: The Universe as a communication network
5: Continuity and Noether's theorem
6: Network layers and the transfinite numbers
7. Logical continuity
8: Can a digital computer network mimic quantum theory?
9: Gravitation: the zero entropy physical network
10: Mathematics as fixed points in the human intellectual layer

Einstein 1954 to Besso: 'I consider it quite possible that physics cannot be based on the field principle, ie on continuous structures. In that case nothing remains of my entire castle in the air, gravitation theory included. Kevin Brown
0. Abstract

Wigner highlights the miraculous correspondence between mathematical symbolism and observations of the physical Universe: . . . the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it. Eugene Wigner

Wigner's observation compels us to examine the relationship between mathematics and physics. Here I wish to suggest an explanation of Wigner's observation. The idea is essentially very simple. Science is devoted to detecting and connecting the fixed points of the Universe. Mathematics, on the other hand, represents the fixed points of a subset of the Universe, the mathematical community.

Insofar as the Universe is one and consistent, it is not surprising that we find a considerable degree of isomorphism or symmetry between these two sets of fixed points. The challenge here is to build a bridge from physics through the mathematical community to the Universe as a whole that clearly illustrates that symmetry. Symmetry - Wikipedia

Quantum mechanics has taught us that all the observable features of the Universe (upon which science is based) are quantized. Nevertheless, quantum mechanics assumes that the mechanism underlying these quantized observations can be represented by continuous mathematics. Here I suggest that this assumption is false, and that we can better describe the Universe by assuming that it is digital 'to the core'.

This digitization suggests that we can see the Universe as a logical, rather than a geometric continuum. The mathematical representation of a logical continuum is the Turing machine, a stepwise digital process that leads deterministically from an initial condition to a final condition. We may see such logical continua as the fixed points in the universal dynamics which form the goal and substance of science. Linear continuum - Wikipedia, Alan Turing, Turing machine - Wikipedia

I guess that the theorems requiring fixed points in a dynamical system are indifferent to the complexity of the system, and I postulate an isomorphism between the dynamics of the mathematical community and the dynamics of the world. I propose that this isomorphism explains the 'unreasonable effectiveness' of mathematics in the sciences. Fixed point (mathematics) - Wikipedia, back to top

1. Introduction: Mathematical theology

This paper is the outcome of a project begun in the 1980s to apply mathematical modelling to theology. This project was motivated by a desire to reduce the religious friction in the world by moving theology toward becoming an evidence based science. The opportunity seemed to be there because both mathematics and theology explore the whole symbolic space bounded by consistency. These bounds have been explored by Cantor, Gödel and Turing and the mathematical work inspired by them. The theological via negativa and mathematical non-constructive proof both assume that divine existence and consistency are equivalent. A Theory of Peace 1987, Constructive proof -Wikipedia, Apophatic theology - Wikipedia, Science - Wikipedia, Georg Cantor - Wikipedia, Kurt Gödel - Wikipedia

Theology is the traditional theory of everything. Early theological works like the Iliad, attributed to Homer. and the Hebrew Bible portray very human gods, but about 500 bce a more scientific strain entered theological thought. Parmenides of Elea asked how we can have certain knowledge in a changing world? His answer, which has remained with us ever since, was to propose a complete, invariant, underlying core to reality which can be truly known. Theology - Wikipedia, Theory of everything - Wikipedia, Homer - Wikipedia, Hebrew Bible - Wikipedia, John Palmer - Parmenides

Theology explicitly entered our scientific history in the Metaphysics of Aristotle. Around 350 bce Aristotle, beginning from his understanding of physics, developed a cosmic vision which became the first steps toward Galileo, Newton and Einstein. Aristotle: Metaphysics

Like Einstein, Aristotle worked from a study of local motion to cosmology. For Einstein, local motion is represented by an inertial frame. Aristotle saw local motion in terms of potentiality and actuality: to move is to change from potentially x to actually x. His theory of potential and actuality has one axiom: no potential can actualize itself. General relativity - Wikipedia, Potentiality and actuality - Wikipedia

Given this axiom, and the fact that we observe motion, he concluded that there must be a first unmoved mover responsible for all the motion in the world. Aristotle placed this mover somewhere in the heavens (made of quintessence) out of human ken, but nevertheless part of the world. Unmoved mover - Wikipedia

Aristotle's work encountered Christian theology in the middle ages in the form of manuscripts transmitted and translated from ancient Greece through Muslim country to Christian Europe. The Catholic Church's premier theologian, Thomas Aquinas, studied Aristotle in William of Moerbecke's literal Latin translation and redeployed Aristotle's proof for the existence of the unmoved mover as a proof for the existence of God. Transmission of the Classics - Wikipedia, Aquinas: Does God exist?

Where Aristotle saw the unmoved mover as part of the Universe, the Judaeo-Christian God existed before the Universe came to be, created it, and micromanages it to this day. Within Christianity, Aquinas' proof is understood to mean that God is not the Universe. There are echoes of the Christian story in the notion that the Universe began with certain fixed initial conditions. Fine-tuned Universe - Wikipedia

Christian theology and cosmology met in the Galielo affair. The outcome of this encounter (to date) has been that science is growing without bounds but Christian theology is essentially unchanged. The problem with Christianity is that the sole source of data for Christian theology is the Bible, and the entrenched belief that Christian theology is essentially true and unchanging. Galileo affair - Wikipedia

The alternative hypothesis, following Aristotle, is to assume that 'everything' does not comprise a god and a universe, but that the Universe itself is divine. Aquinas, following Aristotle, concludes that God is pure activity, actus purus.

There is no potency in God because God is the realization of all possibility. Aquinas then argues from actus purus to absolute simplicity, God is omnino simplex. He then goes on to derive all the traditional properties of the Christian God: infinity, eternity, omnipresence, omniscience, omnipotence, life, intelligence and so on. Attributes of God in Christianity - Wikipedia

The principal problem for me was how can this god be both absolutely simple and omniscient. Absolute simplicity means no marks to carry information, and with no information, no omniscience. Over the years, I have gradually concluded that fixed point theory provides an answer to this problem. Rolf Landauer

There is no contradiction involved in a purely dynamic god having fixed points. The only difference between this and the Christian view is that, following Parmenides, Christian doctrine sees the fixed points as other than the dynamics. Fixed point theory enables us to see that fixed points are simply those points in the dynamics where f(x) = x. back to top

2. Fixed point theory and quantum mechanics.

We consider quantum field theory to be our best attempt so far to produce a comprehensive theory of the physical Universe. Quantum field theory begins with the vacuum, which despite its name is not so much nothing as pure unstructured energy. The vacuum is of itself not observable, but we can model it as an isolated quantum system using the energy equation iℏdψ/dt = Hψ where ψ is a state vector in a Hilbert space of any dimension and H is the energy operator represented by a square matrix of the same dimension as ψ. Quantum field theory - Wikipedia, Vacuum - Wikipedia, Schrödinger equation - Wikipedia

This quantum mechanical model of isolated quantum systems appears, from its predictions, to work perfectly. The solutions to this equation may be a finite, countably infinite or transfinite superposition of states. We may identify the vacuum with the structureless initial singularity postulated by Hawking and Ellis to be the origin of our expanding universe and with the structureless divine source of the Universe proposed by traditional theology. Hawking & Ellis

Since there is nothing outside the vacuum so conceived, we may imagine it as a dynamic system mapping onto itself and so expect it to fulfill the hypotheses of fixed point theory.

Fixed point theory tells us that under various circumstances, dynamic systems can have fixed points. Brouwer's fixed point theorem, for example, says: for any continuous function f mapping a compact convex set into itself there is a point x such that f(x) = x. Brouwer fixed point theorem - Wikipedia

Our expectation of fixed points is met by quantum mechanics. Although it has been often been assumed since ancient times that the Universe is continuous, it is a matter of fact that all our observations are of discrete events. This is true not only at the quantum level, but at all scales where we observe discrete objects like people, leaves or grains of sand. The principal argument for continuity is the apparent continuity of motion. Quantum mechanics - Wikipedia

The underlying mathematical theory suggests that the continuous superposition of solutions to the energy equation evolves deterministically and that each element of the superposition is in perpetual motion at a rate proportional to its energy given by the equation E = hf. The wave equation is normalized so that the sum of all the frequencies to be found in the superposition is equal to the total energy of the system modelled.

Quantum mechanics raised many mathematical questions that were ultimately settled by realizing that quantum mechanics works in a function space, Hilbert space, and that this space is indifferent to the number and length of the vectors considered. Beginning with two-state space, we can work our way up through spaces of countably infinite to transfinite dimensions. The algorithms of quantum mechanics seem to be symmetrical with respect to the dimension of the space in which they operate. von Neumann

An isolated quantum system is observed or measured by coming into contact with another quantum system. An observation is represented by an ‘observable’ or measurement operator, M, and we find that the only states that we see are eigenfunctions of M. These states are the fixed points under the operation of M given by the ‘eigenvalue equation’ Mψ = mψ . The scalar parameter m is the eigenvalue corresponding to the eigenfunction ψ. The eigenfunctions of a measurement operator are M are determinate functions or vectors which can be computed from M. Eigenvalues and eigenvectors - Wikipedia

Although the continuous wave function is believed to evolve deterministically, and the eigenfunctions of a measurement operator can in principle be computed exactly, we can only predict the probability distribution of the eigenvalues revealed by the repetition of a given measurement.

The frequencies are predicted by the Born rule: pk = |<mk | ψ>|2 where ψ is the unknown pre-existing state of the system to be measured and pk is the probability of observing the eigenvalue corresponding to the kth eigenfunction mk of M . Provided the measurement process is properly normalized, the sum of the probabilities pk is 1. When we observe the spectrum of a system, the eigenfunctions determine the frequencies of the lines we observe and the eigenvalues the line weights. Born rule - Wikipedia

The fixed points described by quantum mechanics provide a foundation for all our engineering of stable structures. The purpose of engineering is to manipulate the probability of events in our favour by applying our scientific understanding of how events are constructed in reality.

Brouwer's theorem is topological, relevant to continuous functions. The Katkutani fixed point theorem generalizes Brouwer's theorem to set valued functions. A set valued function may have an infinite set of fixed points. Kakutani fixed-point theorem - Wikipedia, back to top

3. The fixed points of the Universe

Einstein emphasized that the aim of physical science is to determine the invariant features of the Universe, that is its fixed points. This suggests an explanation for Wigner's observation that mathematics often fits the observed Universe with wondrous precision. Both the observable Universe and the mathematics are fixed points in a dynamic system, on the one hand the whole Universe, on the other a subset of the Universe we call the mathematical community. This suggests the existence of a formal symmetry (symmetry with respect to complexity) which couples mathematics to the Universe.

For most of is history, mathematics was confined to the exploration of the magnitudes measured by the natural and real numbers. Classical physics operates in this realm. Georg Cantor opened up a new world when he invented the transfinite numbers. Cantor wanted to find a number big enough to represent the cardinal of the continuum. Transfinite numbers - Wikipedia

He began with the set of natural numbers, cardinal 0, generated the next transfinite cardinal 1 by enumerating the set of all orderings or permutations of the set of natural numbers. 2 is the cardinal of the set of all permutations of the set whose cardinal is 1 and so on without end. Cantor saw the process of permutation as a 'unitary law' which could be used to generate transfinite numbers ad infinitum. Function space - Wikipedia

Let us assume that the transfinite numbers form a space large enough to be placed into one-to-one correspondence with the fixed points of the Universe. Fixed point theorems tell us that certain dynamic systems must have fixed points, and the observations of quantum mechanics give us instances when these theorems apply.

The higher transfinite numbers are very complex objects, being permutations of permutations of . . . and so we can expect to be able to find a transfinite number corresponding to any situation we observe. back to top

4. The Universe as a communication network

We can approach the existence and underlying dynamics of fixed points in the Universe from another direction, by considering the Universe as a communication network. The properties of such a network are defined by the mathematical theory of communication invented by Claude Shannon. The aim of communication is to transmit a true copy of a set of data from one point in space-time to another within the forward light cone of the origin. Claude E Shannon

Shannon develops the theory of communication geometrically by considering a geometrical representation of transmitter and a receiver. The input to the transmitter is a message, a point in message space, and its output is a signal, a point in signal space corresponding to the point in message space. In order to avoid confusion, the transmitter must establish a unique mapping between messages and signals which can be inverted by the receiver. The fundamental strategy for error correction is to make the signal space so large that legitimate messages can be placed so far apart that their probability of confusion is minimal. These messages are, in effect, orthogonal, quantized or digitized.

A quantum measurement may be considered as a communication source. Communication theory characterizes a source S by its alphabet of symbols si and the corresponding probabilities pi of emission of each of the symbols. These probabilities are normalized by the requirement Σi pi = 1, ie the source emits one and only one symbol at a time. The symbols emitted by a quantum measurement are the eigenvalues of the measurement operator and their frequencies, also normalized, are predicted by the Born rule.

The coincidences between the mathematical theory of communication and quantum mechanics suggest that we picture the Universe as a communication network. In this picture, quantum observations or measurements are seen as the transmission of messages between quantum systems.

Freedom from error also requires that the operations of mappings from message to signal and its inverse be deterministic, that is the mappings must be computable functions. This requirement supports the guess that the total set of eigenfunctions of the Universe is the set of computable functions and so is equivalent to the first transfinite cardinal0. Computable function - Wikipedia

This identification is equivalent to the quantum mechanical trick of placing the system under study in a finite box to select a finite number of states. The box here is the set of computable functions which we propose to form the computational foundation of a network spanning the space of fixed points in the Universe. back to top

5. Continuity and Noether's theorem

Even when talking about continuous quantities, mathematics is expressed in symbolic or digital form. Physical motion appears continuous and so it has been accepted since ancient times the the space and time in which we observe motion are also continuous. From the point of view of communication theory and algorithmic information theory, a continuum, with no marks or modulation, can carry no information. This idea is supported by Emmy Noether's theorem which links symmetries to invariances and conservation laws, different expressions of the fixed points in the dynamic Universe. Neuenschwander, Nina Byers, Noether's theorem - Wikipedia

Each of these three terms is equivalent to the statement 'no observable motion'. From the observers point of view nothing happens, although we may imagine and model some invisible motion or transformation like the rotation of a perfect sphere. Gauge theory - Wikipedia

The Universe is believed to have evolved from a structureless initial singularity to its current state. The evolutionary process is a product of variation and selection. The variation is made possible by the existence of non-deterministic (ie non-computable) processes. The duplication of genetic material during cell division, for instance, is subject to a certain small error rate which may ultimately affect the fate of the daughter cells. Evolution - Wikipedia

Selection culls the variations. The net effect of variation and selection is to optimize systems for survival, that is for stability or the occupation of a fixed point (which may be in a space of transfinite dimension). Before the explicit modelling of evolution, however, writers like de Maupertuis and others speculated that the processes of the world were as perfect as possible. Yourgrau & Mandelstam: Variational principles in dynamics and quantum theory

Mathematical physics eventually captured this feeling using Lagrangian mechanics. An important result of this search is Hamilton’s principle: that the world appears to optimize itself using a principle of stationary action. Noether succeeded in coupling the action functional to invariance and symmetry, to give us a broad picture of the bounds on the Universe as those fixed points where nothing happens. The conservation of action (angular momentum) , energy, and momentum form the backbone of modern physics.

Noether's work is based on continuous transformations represented by Lie groups. Symmetry also applies to discrete transformations, as we can see by rotating a triangle or a snowflake. We understand symmetries by using the theory of probability. We may consider all the 'points' in a continuous symmetry as equiprobable, and for some discrete symmetries this is also true as we see in a fair coin or an unloaded die (ignoring the identifying marks on the faces). Lie Group - Wikipedia

Communication theory also introduces the statistics of a communication source whose discrete letters are not equiprobable, but each letter of the alphabet ai has probability pi such that Σi pi = 1. Shannon then defines a source entropy H = -Σi pi log pi, which is maximized when the pi are all equal. We may consider a quantum system as a communication source emitting discrete letters events with a certain probability structure. The condition Σi pi = 1 is enforced in quantum mechanics by normalization.

From this point of view, quantum mechanics predicts the frequency of traffic on different legs of the universal network and quantum field theory enables us to model the nature and behaviour of the messages (particles) passing through this network. back to top

6. Network layers and the transfinite numbers

Engineered networks are layered, a technology necessary to make them easy to construct, expand and troubleshoot. It has long been noticed that the world itself is layered, larger events being built out of smaller ones until we come to an ultimate atom of action measured by Planck's constant, h. Tanenbaum: Computer Networks

We transform this idea into a transfinite network by mapping the layers of the universal network onto the sequence transfinite numbers, beginning by letting the natural numbers correspond to the physical layer of the Universe. The eigenfunctions of this physical layer are the countably infinite set of Turing machines.

Each subsequent software layer uses the layer beneath it as an alphabet of operations to achieve its ends. The topmost layer, in engineered networks, comprises human users. These people may be a part of a corporate network, reporting through further layers of management to the board of an organization.

By analogy to this layered hierarchy, we may consider the Universe as a whole as the ultimate user of the universal network. Since the higher layers depend on the lower layers for their existence, we can expect an evolutionary tendency for higher layers to curate their alphabets to maintain its own stability.

Processes in corresponding layers (‘peers’) of two nodes in a network may communicate if they share a suitable protocol. All such communication uses the services of all layers between the peers and the physical layer. These services are generally invisible or transparent to the peers unless they fail. Thus two people in conversation are generally unaware of the huge psychological, physiological and physical complexity of the systems that make their communication possible.

Let us imagine that the actual work of permutation in the symmetric universe (ie its dynamics) is executed by Turing machines. As formal structures these Turing machines are themselves ordered sets, and are to be found among the ordered strings contained in the Universe.

The installation of these Turing machines turns the transfinite universe into the transfinite network. This network is a set of independent memories able to communicate with and change one another via Turing machines. The internet is a finite example of such a network, the memories of servers, routers, clients and users changing each other’s states through communication.

It seems clear that the transfinite network has sufficient variety to be placed in one-to-one correspondence with any structure or process in the Universe. In a case where a given layer of the network universe is found to be too small to accommodate the system of interest, we have only to move up through the layers until we find a level whose cardinal is adequate for the task. Ashby Cybernetics

Permutations can be divided into subsets or cycles of smaller closed permutations. This process means that no matter what the cardinal of a permutation, we can find finite local permutations whose action nevertheless permutes the whole Universe. Moving my pen from a to b (and moving an equivalent volume of air from b to a) is such an action. Permutation - Wikipedia

Although there are 1 mappings of the 0 natural numbers onto themselves, there are only 0 different Turing machines. As a consequence, almost all mappings are incomputable, and so cannot be generated by a deterministic process. Nevertheless a mapping once discovered may be tested by a computable process. Here we see an echo of the P versus NP problem. P versus NP problem - Wikipedia

From a communication point of view, quantum mechanics does not reveal actual messages but rather the traffic on various links. If we assume that the transmission of a message corresponds to a quantum of action, the rate of transmission in a channel is equivalent to the energy on that channel, information encoded in the energy operator, H.

Further, the collapse of the wave function may be analogous to the completion of a halting computation. The completion of a computation is associated with a quantum of action. Eigenfunctions are orthogonal to one another to prevent error. Every eigenfunction has an inverse to decode the message it has encoded. Wave function collapse - Wikipedia, back to top

7: Logical continuity

Noether's work depends on analytic continuity. Mathematics as a whole, however, depends upon logical continuity. A logical continuum is a halting Turing machine which proceeds by logical steps from an initial to a final state. The structure of mathematics is built up by conjecture and proof. Each proof is in effect a Turing machine leading deterministically from hypotheses to conclusions. Continuity - Wikipedia, Mathematical proof - Wikipedia

We may view an analytic continuum as the carrier of either no information, or an infinite amount of information. From an engineering point of view a continuum carries no information other than that it is present. From the point set point of view, however, a continuum contains a transfinite number of points, each of which can be used as a mark for the purpose of representing information.

The hope that a quantum computer may be more powerful than a Turing machine depends upon the latter view, but it may be that the Universe itself takes the engineering approach. The search for underlying analytic continuity is pointless if a continuum is not observable. This idea is consistent with Landauer's idea that information is physical.

What we observe is that networks based on computers are able to send messages error free through noisy environments, and that the existence of logical continuity in the Universe makes the existence of complex ordered structures possible. back to top

8. Can a digital computer network mimic quantum theory?

Quantum mechanics as we know it is based on continuous (that is analogue) computation. Can a digital computer produce the same results as quantum mechanics? In other words, is the Universe digital 'to the core', founded on logical rather than analytic continuity? From an algorithmic point of view, continuous symmetries represent nothing, that is they are computationally equivalent to no-operation. They are the boundaries of the observable Universe, equivalent in communication terms to no signal, only (perhaps) an unmodulated carrier. Carrier signal - Wikipedia

Like Parmenides, quantum theory proposes an invisible deterministic process underlying the observed world. Traditionally this invisible process is modelled by continuous mathematics. It seems that a digital computation would also be invisible. To transmit its instantaneous internal state, a computer must stop what it is doing and process a message to the outside observer. It is impossible for a computer to transmit every state of its operation to an observer because the transmission of messages is also a computation and the process would never halt.

Since Feynman devised quantum Hamiltonians that modelled the classical functions of logic, there has been a growing conviction that quantum processes may be modelled as computations. Feynman: Lectures on Computation

Quantum mechanics is based on the field of complex numbers. Complex numbers are essential to quantum mechanics first because they are periodic (and so can model the clock and all the other periodic processes in a computer), and second because of their arithmetic properties: we model interference by addition and changing phase (motion in space-time) by multiplication. These two features are combined in Feynman's path integral method to yield a fixed amplitude for various quantum processes. Complex number - Wikipedia, Path integral formulation - Wikipedia

A complex number has two orthogonal dimensions which communicate by multiplication, i2 = −1. There is no problem implementing finite versions of complex arithmetic in a digital computer.

There are high hopes in the quantum computing community that we may eventually devise quantum mechanical computers more powerful than Turing machines. The essence of quantum computation's claim to greater power is that a formally perfect analogue machine can transform large sets of data (ie representations of real or complex numbers) in one operation. This assumption implies that state vectors can carry an infinite amount of information and that matrix operations on these vectors are in effect massively parallel computations, dealing with the complete basis of the relevant Hilbert space simultaneously.

The atomic process  of a digital computer, on the other hand, is a one bit operation, p becomes not-p. However the logical proof  of the analogue contention is digital, using point set theory. Point set theory assumes that all points in a continuum are orthogonal and uniquely addressed by real numbers.

The epsilons and deltas in Weierstrasse's formal definition of continuity are at every point in the limiting process definite numbers. As we approach the continuous limit, these numbers are believed to hold a definite functional relationship to one another even as their measures approach zero. From an algorithmic points of view, the strength of this argument lies in the assumption that this functional relationship holds. We find in the physical world, however, that there are no discrete symbols of measure zero: the smallest meaningful measure is Planck's constant. Karl Weierstrass - Wikipedia, Algorithmic information theory - Wikipedia

Cantor explicitly quantized the study of the continuum by inventing set theory which deals with 'definite and separate objects'. Cantor set out to measure the cardinal of the continuum using set theory. Cohen later showed this is not possible, since the concept of set is independent of (orthogonal to) cardinality, ie sets are symmetrical with respect to size, so that no information about a cardinal is available from purely set theoretical considerations. Cantor, Cohen

The formalism of quantum mechanics enjoys a similar symmetry: it is indifferent to the number of components in its vectors, that is to the dimension of the Hilbert space of interest. We accept systems from one state up to the cardinal of the continuum where the quantum formalism is used to represent a classically continuous variables. We this property is also a symmetry with respect to complexity and serves as bridge to connect Hilbert spaces with any number of dimensions.

Logical symmetry also enjoys symmetry with respect to complexity, so that logical arguments about large and complex sets obey the same rules as logical arguments about atomic entities. Logical continuity (epitomized by current cosmology) thus carries us from the initial state of the Universe to its current state, and gives us means to study the future. Algorithmic information theory - Wikipedia, back to top

9: Gravitation: the zero entropy physical network

Here we understand the transfinite universe as layered set of function spaces generated by permutation. The cardinal of the set of mappings of a set of 0 symbols onto itself is 1 and so on without end.This structure is capable of representing any group, since the permutation group of any finite cardinality contains all possible groups of that power.

In an engineered network all messages between users drill down through the software layers to the physical layer at the transmitting end. At the receiving end the message works its way up through layers of software to the receiving user. How does this process look in the physical world?

The general structure of the Universe as we know it is layered in various ways. Following this trail backwards, we come to the initial singularity, which we assume to be pure action primed to differentiate into the current Universe.

We imagine gravitation to describe the layer of the Universal network which is concerned with the transmission of meaningless identical symbols, simply quanta of action. Gravitation sees only undifferentiated energy, and is therefore blind to the higher layers of the Universe which introduce memory, correspondences and meaning. Quantum mechanics is also non-local, suggesting that it describes a layer of the universal network antecedent to the spatial layer. This view is reinforced by the observation that quantum mechanics requires no memory, each state deriving from the state immediately preceding it. back to top

10: Mathematics as fixed points in the human intellectual layer

At some point in the transfinite hierarchy we come to the human layer, the layer in which the peers are human beings. It seems clear enough that a communication model of the Universe fits the human world quite well.

We may interpret written mathematics (like Cantor’s papers) as a representation of fixed points within the mathematical community insofar as mathematics is archived and communicated through space and time by the literature. As a representation it is identical to the particles that we observe when we study the physical Universe and we may take the view that the creation of both literature and physics are isomorphic processes differing only in complexity. We map this complexity using the Cantor hierarchy of transfinite Hilbert spaces and assume that they are all share the property that they are logically consistent structures.

This isomorphism, I suggest, explains the amazing utility of mathematics as a language to describe the Universe. back to top

(revised 19 June 2016)

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 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.' 
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Cohen, Paul J, Set Theory and the Continuum Hypothesis, Benjamin/Cummings 1966-1980 Preface: 'The notes that follow are based on a course given at Harvard University, Spring 1965. The main objective was to give the proof of the independence of the continuum hypothesis [from the Zermelo-Fraenkel axioms for set theory with the axiom of choice included]. To keep the course as self contained as possible we included background materials in logic and axiomatic set theory as well as an account of Gödel's proof of the consistency of the continuum hypothesis. . . .'  
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Feynman, Richard, Feynman Lectures on Computation, Perseus Publishing 2007 Amazon Editorial Reviews Book Description 'The famous physicist's timeless lectures on the promise and limitations of computers When, in 1984-86, Richard P. Feynman gave his famous course on computation at the California Institute of Technology, he asked Tony Hey to adapt his lecture notes into a book. Although led by Feynman, the course also featured, as occasional guest speakers, some of the most brilliant men in science at that time, including Marvin Minsky, Charles Bennett, and John Hopfield. Although the lectures are now thirteen years old, most of the material is timeless and presents a "Feynmanesque" overview of many standard and some not-so-standard topics in computer science such as reversible logic gates and quantum computers.'  
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Hawking, Steven W, and G F R Ellis, The Large Scale Structure of Space-Time , Cambridge UP 1975 Preface: Einstein's General Theory of Relativity . . . leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.' 
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Neuenschwander, Dwight E, Emmy Noether's Wonderful Theorem, Johns Hopkins University Press 2011 Jacket: A beautiful piece of mathematics, Noether's therem touches on every aspect of physics. Emmy Noether proved her theorem in 1915 and published it in 1918. This profound concept demonstrates the connection between conservation laws and symmetries. For instance, the theorem shows that a system invariant under translations of time, space or rotation will obey the laws of conservation of energy, linear momentum or angular momentum respectively. This exciting result offers a rich unifying principle for all of physics.' 
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Tanenbaum, Andrew S, Computer Networks, Prentice Hall International 1996 Preface: 'The key to designing a computer network was first enunciated by Julius Caesar: Divide and Conquer. The idea is to design a network as a sequence of layers, or abstract machines, each one based upon the previous one. . . . This book uses a model in which networks are divided into seven layers. The structure of the book follows the structure of the model to a considerable extent.'  
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von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '. . . a revolutionary book that caused a sea change in theoretical physics. . . . JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.' 
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Yourgrau, Wolfgang, and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory, Dover 1979 Variational principles serve as filters for parititioning the set of dynamic possibilities of a system into a high probability and a low probability set. The method derives from De Maupertuis (1698-1759) who formulated the principle of least action, which states that physical laws include a rule of economy, the principle of least action. This principle states that in a mathematically described dynamic system will move so as to minimise action. Yourgrau and andelstam explains the application of this principle to a variety of physical systems.  
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Links
Actus purus - Wikipedia, Actus purus - Wikipedia, the free encyclopedia, 'Actus Purus is a term employed in scholastic philosophy to express the absolute perfection of God. It literally means, "pure act." Created beings have potentiality that is not actuality, imperfections as well as perfection. Only God is simultaneously all that He can be, infinitely real and infinitely perfect: `I am who I am`(Exodus 3:14). His attributes or His operations, are really identical with His essence, and His essence includes essentially His existence.' back
Alan Turing, On Computable Numbers, with an application to the Entscheidungsproblem, 'The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique.' back
Algorithmic information theory - Wikipedia, Algorithmic information theory - Wikipedia, the free encyclopedia, 'Algorithmic information theory is a subfield of information theory and computer science that concerns itself with the relationship between computation and information. According to Gregory Chaitin, it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously."' back
Apophatic theology - Wikipedia, Apophatic theology - Wikipedia, the free encyclopedia, 'Apophatic theology (from Greek ἀπόφασις from ἀπόφημι - apophēmi, "to deny")—also known as negative theology or via negativa (Latin for "negative way")—is a theology that attempts to describe God, the Divine Good, by negation, to speak only in terms of what may not be said about the perfect goodness that is God. It stands in contrast with cataphatic theology.' back
Aquinas 13, Summa: I 2 3: Whether God exists?, I answer that the existence of God can be proved in five ways. The first and more manifest way is the argument from motion. . . . The second way is from the nature of the efficient cause. . . . The third way is taken from possibility and necessity . . . The fourth way is taken from the gradation to be found in things. . . . The fifth way is taken from the governance of the world. back
Aristotle, Metaphysics, 'Written 350 B.C.E, Translated by W. D. Ross. Book I Part 1 "ALL men by nature desire to know. An indication of this is the delight we take in our senses; for even apart from their usefulness they are loved for themselves; and above all others the sense of sight. For not only with a view to action, but even when we are not going to do anything, we prefer seeing (one might say) to everything else. The reason is that this, most of all the senses, makes us know and brings to light many differences between things. ' back
Attributes of God in Christianity - Wikipedia, Attributes of God in Christianity - Wikipedia, the free encyclopedia, 'The attributes of God are specific characteristics of God discussed in Christian theology.' back
Born rule - Wikipedia, Born rule - Wikipedia, the free encyclopedia, 'The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see bra-ket notation), then the measured result will be one of the eigenvalues λ of A, and the probability of measuring a given eigenvalue λi will equal <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'. back
Brouwer fixed point theorem - Wikipedia, Brouwer fixed point theorem - Wikipedia, the free encyclopedia, 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself. back
Carrier signal - Wikipedia, Carrier signal - Wikipedia, the free encyclopedia, 'In telecommunications, a carrier signal, carrier wave, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an input signal for the purpose of conveying information. This carrier wave is usually a much higher frequency than the input signal. The purpose of the carrier is usually either to transmit the information through space as an electromagnetic wave (as in radio communication), or to allow several carriers at different frequencies to share a common physical transmission medium by frequency division multiplexing (as, for example, a cable television system). The term is also used for an unmodulated emission in the absence of any modulating signal.' back
Claude E Shannon, A Mathematical Theory of Communication, 'The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.' back
Complex number - Wikipedia, Complex number - Wikipedia, the free encyclopedia, 'IA complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane (also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part.' back
Computable function - Wikipedia, Computable function - Wikipedia, the free encyclopedia, 'Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. They make precise the intuitive notion of algorithm. Computable functions can be used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Their definition, however, must make reference to some specific model of computation.' back
Constructive proof -Wikipedia, Constructive proof -Wikipedia, the free encyclopedia, 'In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem) which proves the existence of a particular kind of object without providing an example.' back
Continuity - Wikipedia, Continuity - Wikipedia, the free encyclopedia, 'In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.' back
Eigenvalues and eigenvectors - Wikipedia, Eigenvalues and eigenvectors - Wikipedia, the free encyclopedia, 'An eigenvector of a square matrix A is a non-zero vector vthat, when the matrix multiplies yields a constant multiple of v, the latter multiplier being commonly denoted by λ. That is: Av = λv' back
Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, 'The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.' back
Evolution - Wikipedia, Evolution - Wikipedia, the free encyclopedia, '. . . Charles Darwin and Alfred Wallace were the first to formulate a scientific argument for the theory of evolution by means of natural selection. Evolution by natural selection is a process that is inferred from three facts about populations: 1) more offspring are produced than can possibly survive, 2) traits vary among individuals, leading to different rates of survival and reproduction, and 3) trait differences are heritable. . . . ' back
Fine-tuned Universe - Wikipedia, Fine-tuned Universe - Wikipedia, the free encyclopedia, 'The fine-tuned Universe is the proposition that the conditions that allow life in the Universe can only occur when certain universal fundamental physical constants lie within a very narrow range, so that if any of several fundamental constants were only slightly different, the Universe would be unlikely to be conducive to the establishment and development of matter, astronomical structures, elemental diversity, or life as it is understood. The proposition is discussed among philosophers, scientists, theologians, and proponents and detractors of creationism.' back
Fixed point (mathematics) - Wikipedia, Fixed point (mathematics) - Wikipedia, the free encyclopedia, 'In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. That is to say, x is a fixed point of the function f if and only if f(x) = x' back
Function space - Wikipedia, Function space - Wikipedia, the free encyclopedia, 'In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both' back
Galileo affair - Wikipedia, Galileo affair - Wikipedia, the free encyclopedia, 'The Galileo affair was a sequence of events, beginning around 1610, during which Galileo Galilei came into conflict with both the Catholic Church, for his support of Copernican astronomy, and secular philosophers, for his criticism of Aristotelianism.' back
Gauge theory - Wikipedia, Gauge theory - Wikipedia, the free encyclopedia, 'In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations. . . . The term gauge refers to redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field.' back
General relativity - Wikipedia, General relativity - Wikipedia, the free encyclopedia, 'General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916.[1] It is the current description of gravitation in modern physics. General relativity generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.' back
Georg Cantor - Wikipedia, Georg Cantor - Wikipedia, the free encyclopedia, Georg Ferdinand Ludwig Philipp Cantor (March 3 [O.S. February 19] 1845[1] – January 6, 1918) was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware' back
Hebrew Bible - Wikipedia, Hebrew Bible - Wikipedia, The Hebrew Bible . . . is a term referring to the books of the Jewish Bible as originally written mostly in Biblical Hebrew with some Biblical Aramaic. The term closely corresponds to contents of the Jewish Tanakh and the Protestant Old Testament (see also Judeo-Christian) but does not include the deuterocanonical portions of the Roman Catholic or the Anagignoskomena portions of the Eastern Orthodox Old Testaments. The term does not imply naming, numbering or ordering of books, which varies (see also Biblical canon).' back
Homer - Wikipedia, Homer - Wikipedia, the free encyclopedia, 'In the Western classical tradition, Homer (. . . Ancient Greek: Ὅμηρος [hómɛːros], Hómēros) is the author of the Iliad and the Odyssey, and is revered as the greatest of ancient Greek epic poets. These epics lie at the beginning of the Western canon of literature, and have had an enormous influence on the history of literature. When he lived is unknown. Herodotus estimates that Homer lived 400 years before his own time, which would place him at around 850 BC, while other ancient sources claim that he lived much nearer to the supposed time of the Trojan War, in the early 12th century BC. Modern researchers appear to place Homer in the 7th or 8th centuries BC.' back
Internet - Wikipedia, Internet - Wikipedia, the free encyclopedia, 'The Internet is a global system of interconnected computer networks that use the standard Internet protocol suite (TCP/IP) to link several billion devices worldwide. It is an international network of networks that consists of millions of private, public, academic, business, and government packet switched networks, linked by a broad array of electronic, wireless, and optical networking technologies.' back
John Palmer - Parmenides, Parmenides (Stanford Encyclopedia of Philosophy), First published Fri Feb 8, 2008 ' Immediately after welcoming Parmenides to her abode, the goddess describes as follows the content of the revelation he is about to receive: You must needs learn all things,/ both the unshaken heart of well-rounded reality/ and the notions of mortals, in which there is no genuine trustworthiness./ Nonetheless these things too will you learn, how what they resolved/ had actually to be, all through all pervading. (Fr. 1.28b-32) ' back
Kakutani fixed-point theorem - Wikipedia, Kakutani fixed-point theorem - Wikipedia, the free encyclopedia, 'In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.' back
Karl Weierstrass - Wikipedia, Karl Weierstrass - Wikipedia. the free encyclopedia, 'Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the "father of modern analysis".' back
Kevin Brown, Reflections on Relativity, To Besso in 1954, nearly 50 years after their discussion in the patent office, Einstein wrote: I consider it quite possible that physics cannot be based on the field principle, i.e., on continuous structures. In that case, nothing remains of my entire castle in the air, gravitation theory included..."' back
Kurt Gödel - Wikipedia, Kurt Gödel - Wikipedia, the free encyclopedia, 'Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.' back
Lie Group - Wikipedia, Lie Group - Wikipedia, the free encyclopedia, 'In mathematics, a Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. . . . Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.' back
Linear continuum - Wikipedia, Linear continuum - Wikipedia, the free encyclopedia, 'In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound.' back
Mathematical proof - Wikipedia, Mathematical proof - Wikipedia, the free encyclopedia, 'In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to generally accepted statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproven statement that is believed true is known as a conjecture.' back
Nina Byers, E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws , Abstract: Emmy Noether proved two deep theorems, and their converses, on the connection between symmetries and conservation laws. ... The present paper in an historical account of the circumstances in which she discovered and proved these theorems which physicists refer to collectively as Noether's Theorem. The work was done soon after Hilbert's discovery of the variational principle which gives the field equations of general relativity. The failure of local energy conservation was a problem that concerned people at that time, among them David Hilbert, Felix Klein and Albert Einstein. Noether's theorems solved this problem. With her characteristically deep insight and through analysis, in solving the problem she discovered very general theorems that have profoundly influenced modern physics. back
Noether's theorem - Wikipedia, Noether's theorem - Wikipedia, the free encyclopedia, 'Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.' back
P versus NP problem - Wikipedia, P versus NP problem - Wikipedia, the free encyclopedia, 'The P versus NP problem is a major unsolved problem in computer science. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. It was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" and is considered by many to be the most important open problem in the field. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution.' back
Path integral formulation - Wikipedia, Path integral formulation - Wikipedia, the free encyclopedia, 'The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude. . . . This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970s which unified quantum field theory with statistical mechanics. . . . ' back
Permutation - Wikipedia, Permutation - Wikipedia, the free encyclopedia, 'In mathematics, the notion of permutation relates to the act of permuting, or rearranging, members of a set into a particular sequence or order (unlike combinations, which are selections that disregard order). For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. The study of permutations of finite sets is a topic in the field of combinatorics.' back
Potentiality and actuality - Wikipedia, Potentiality and actuality - Wikipedia, the free encyclopedia, 'In philosophy, Potentiality and Actualit are principles of a dichotomy which Aristotle used to analyze motion, causality, ethics, and physiology in his Physics, Metaphysics, Ethics and De Anima (which is about the human psyche). The concept of potentiality, in this context, generally refers to any "possibility" that a thing can be said to have. Aristotle did not consider all possibilities the same, and emphasized the importance of those that become real of their own accord when conditions are right and nothing stops them.[3] Actuality, in contrast to potentiality, is the motion, change or activity that represents an exercise or fulfillment of a possibility, when a possibility becomes real in the fullest sense. 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 field theory - Wikipedia, Quantum field theory - Wikipedia, the free encyclopedia, 'Quantum field theory (QFT) provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or (especially in a condensed matter context) of many-body systems. . . . In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta - necessarily chunked ripples in a field that 'look like' particles. Fermions, like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta). back
Quantum mechanics - Wikipedia, Quantum mechanics - Wikipedia, the free encyclopedia, 'Quantum mechanics (QM; also known as quantum physics or quantum theory), including quantum field theory, is a fundamental branch of physics concerned with processes involving, for example, atoms and photons. In such processes, said to be quantized, the action has been observed to be only in integer multiples of the Planck constant. This is utterly inexplicable in classical physics.'' back
Rolf Landauer, Information is a Physical Entity, 'Abstract: This paper, associated with a broader conference talk on the fundamental physical limits of information handling, emphasizes the aspects still least appreciated. Information is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possibilities of our real physical universe. The mathematician's vision of an unlimited sequence of totally reliable operations is unlikely to be implementable in this real universe. Speculative remarks about the possible impact of that, on the ultimate nature of the laws of physics are included.' back
Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, 'IIn quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back
Science - Wikipedia, Science - Wikipedia, the free encyclopedia, 'Science (from Latin scientia, meaning "knowledge") is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. In an older and closely related meaning, "science" also refers to a body of knowledge itself, of the type that can be rationally explained and reliably applied. A practitioner of science is known as a scientist..' back
Symmetry - Wikipedia, Symmetry - Wikipedia, the free encyclopedia, 'Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to a transformation, such as reflection but including other transforms too. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together.' back
Theology - Wikipedia, Theology - Wikipedia, the free encyclopedia, Theology is the systematic and rational study of concepts of God and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university, seminary or school of divinity. . . . 'During the High Middle Ages, theology was therefore the ultimate subject at universities, being named "The Queen of the Sciences" and serving as the capstone to the Trivium and Quadrivium that young men were expected to study. This meant that the other subjects (including Philosophy) existed primarily to help with theological thought.' back
Theory of everything - Wikipedia, Theory of everything - Wikipedia, the free encyclopedia, 'A theory of everything (ToE) or final theory, ultimate theory, or master theory refers to the hypothetical presence of a single, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe. ToE is one of the major unsolved problems in physics. Over the past few centuries, two theoretical frameworks have been developed that, as a whole, most closely resemble a ToE. The two theories upon which all modern physics rests are General Relativity (GR) and Quantum Field Theory (QFT). ' back
Transfinite numbers - Wikipedia, Transfinite numbers - Wikipedia, the free encyclopedia, 'Transfinite numbers are cardinal numbers or ordinal numbers that 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 workers 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
Transmission of the Classics - Wikipedia, Transmission of the Classics - Wikipedia, the free encyclopedia, 'The introduction of Greek philosophy and science into the culture of the Latin West in the Middle Ages was an event that transformed the intellectual life of Western Europe. It consisted of the discovery of many original works, such as those written by Aristotle in the classical period. Greek manuscripts have been maintained in the Greek speaking world in Constantinople, Armenia, Syria, and Alexandria. Interest and availability of Greek text was scarce in the Latin West until with increase traffic to the East, including the Latin Empire during the time of the Crusade, the Sack of Constantinople during the 4th Crusade, and finally the conquest of Constantinople by the Ottoman Empire caused many of the original Greek manuscripts to make their way into Western Europe, and thus fueled the Renaissance.' back
Turing machine - Wikipedia, Turing machine - Wikipedia, the free encyclopedia, A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a computer. The "machine" was invented in 1936 by Alan Turingwho called it an "a-machine" (automatic machine). The Turing machine is not intended as practical computing technology, but rather as a hypothetical device representing a computing machine. Turing machines help computer scientists understand the limits of mechanical computation.' back
Unmoved mover - Wikipedia, Unmoved mover - Wikipedia, the free encyclopedia, 'The unmoved mover (ού κινούμενον κινεῖ oú kinoúmenon kineῖ) is a philosophical concept described by Aristotle as a primary cause or "mover" of all the motion in the universe. As is implicit in the name, the "unmoved mover" is not moved by any prior action. In Book 12 (Greek "Λ") of his Metaphysics, Aristotle describes the unmoved mover as being perfectly beautiful, indivisible, and contemplating only the perfect contemplation: itself contemplating. He equates this concept also with the Active Intellect. This Aristotelian concept had its roots in cosmological speculations of the earliest Greek "Pre-Socratic" philosophers and became highly influential and widely drawn upon in medieval philosophy and theology. St. Thomas Aquinas, for example, elaborated on the Unmoved Mover in the quinque viae.' back
Vacuum - Wikipedia, Vacuum - Wikipedia, the free encyclopedia, 'According to modern understanding, even if all matter could be removed from a volume, it would still not be "empty" due to vacuum fluctuations, dark energy, transiting gamma- and cosmic rays, neutrinos, along with other phenomena in quantum physics. In modern particle physics, the vacuum state is considered as the ground state of matter.' back
Wave function collapse - Wikipedia, Wave function collapse - Wikipedia, the free encyclopedia, 'In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several eigenstates—appears to reduce to a single eigenstate (by "observation"). It is the essence of measurement in quantum mechanics, and connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is continuous evolution via the Schrödinger equation.' back

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