##### vol **6:** Essays

### The unreasonable effectiveness of mathematics revisited

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: * p _{k}* = |<

*m*|

_{k }*ψ*>|

^{2}where

*ψ*is the unknown pre-existing state of the system to be measured and

*p*is the probability of observing the eigenvalue corresponding to the

_{k}*k*th eigenfunction

*m*of

_{k}*M*. Provided the measurement process is properly normalized, the sum of the probabilities

*p*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

_{k}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 *ℵ _{0}*, generated the next transfinite cardinal

*ℵ*by enumerating the set of all orderings or permutations of the set of natural numbers.

_{1}*ℵ*is the cardinal of the set of all permutations of the set whose cardinal is

_{2}*ℵ*and so on without end. Cantor saw the process of permutation as a 'unitary law' which could be used to generate transfinite numbers

_{1}*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 *s _{i}* and the corresponding probabilities

*p*of emission of each of the symbols. These probabilities are normalized by the requirement

_{i}*Σ*= 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.

_{i}p_{i}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 cardinal* ℵ _{0}*. 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 *p _{i}* such that

*Σ*= 1. Shannon then defines a source entropy

_{i}p_{i }*H = -Σ*, which is maximized when the

_{i}p_{i}log p_{i}*p*are all equal. We may consider a quantum system as a communication source emitting discrete letters events with a certain probability structure. The condition

_{i}*Σ*= 1 is enforced in quantum mechanics by normalization.

_{i}p_{i}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 *ℵ _{1}* mappings of the

*ℵ*natural numbers onto themselves, there are only

_{0}*ℵ*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

_{0}
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,* i ^{2}* = −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*ℵ*symbols onto itself is

_{0}*ℵ*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.

_{1}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)