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vol 6: Essays

Why do we observe a quantized Universe?

1. Abstract
2. Computer networks
3. The mathematical theory of communication
4. Turing machines and ‘logical continuity’
5. Communication and quantum mechanics
6. Coding delay and special relativity
7. ‘Spooky action at a distance’
8. Local gauge invariance in a network
9. The transfinite network
10. Formalism and the physical world
11. Creation
12. Annihilation
13. Determinism and indeterminism
14.The distinction between past and future
15. Quantum computation
16. Evolution
17. Conclusion: ‘The unreasonable effectiveness of mathematics’
1. Abstract

Why do we observe a quantized Universe? Here I propose that it is because we and every other entity in the Universe are parts of a digital computer network. We begin to model the Universe as a finite computer network like the internet Tanenbaum. We extend this model mathematically to a network with a countable infinity of fundamental processes corresponding to the computable functions represented by halting Turing machines Alan Turing. Mathematical continuity is replaced by the more powerful notion of logical continuity implemented formally by mathematical proof and practically by symbolic computing machines.

2. Computer networks

Since all information is encoded physically, practical computer networks are built of a physical layer which correlates physical states or signals with the information in messages Rolf Landauer. This hardware layer is driven by various strata of software. A stable network requires error free communication so that the first software layer in practical networks is usually devoted to error detection and correction.

An ‘atomic’ communication is represented by the transmission of single packet from one source to another. Practical point to point communication networks connect many sources, all of which are assigned addresses so that addressed packets may be steered to their proper recipient. This ‘post office’ work is implemented by further network layers.

Each subsequent software layer uses the layer beneath it as an alphabet of operations to achieve its ends. The topmost layer, in computer 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.

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.

3. The mathematical theory of communication

The mathematical theory of communication shows that we can make communication error free by coding our messages into packets that are so far apart in message space that the probability of their confusion is negligible. Shannon sought the limits of error free communication over noiseless and noisy channels. The theory he developed is now well known and lies at the heart of communication networks worldwide Claude Shannon, Claude E Shannon, Khinchin

The validity of these strategies is illustrated by our current ability to send gigabytes of information error free over noisy phone lines. The quantization of communication at the microscopic level supports the hypothesis that our world is a communication network that has evolved to resist error Wojciech Hubert Zurek.

A system that transmits without errors at the limiting rate C predicted by Shannon’s theorems is called an ideal system. Some features of an ideal system are:

1. To avoid error there must be no overlap between signals representing different messages, They must, in other words, be orthogonal, as with the eigenfunctions of a quantum mechanical basis.

2. Such ‘basis signals’ may be chosen at random in the signal space, provided only that they are orthogonal. The same message may be encoded into any satisfactory basis provided that the transformations used by used by the transmitter and receiver to encode the message into the signal and decode the signal back to the message are inverses of one another.

3. The signals transmitted by an ideal system are indistinguishable from noise. The fact that a set of physical observations looks like a random sequence is not therefore evidence for meaninglessness. Until the algorithms used to encode and decode such a sequence are known, nothing can be said about its significance.

4. Only in the simplest cases are the mappings used to encode and decode messages linear and topological. For practical purposes, however, they must all be computable with available machines.

5. As a system approaches the ideal, the length of the transmitted packets, the delay at the transmitter while it takes in a chunk of message for encoding, and the corresponding delay at the receiver, increase indefinitely.

4. Turing machines and ‘Logical Continuity’

Classical physics, from the time of Aristotle, has assumed that the physical Universe is continuous Aristotle. Democritus, it is true, postulated discrete atoms, but even these moved in a continuous space. This is a natural assumption, given the continuous appearance of macroscopic spatial motion.

Yet, as a matter of fact, all observations and discussions of the Universe are quantized. Experimental physics revolves around classifying (‘binning’) and counting events. When we observe the output of physicists and mathematicians (‘the literature’) we see that it too is quantized, into discrete volumes, articles, words and symbols, like this.

Shannon showed that appropriate coding enables free communication but his work did not reveal the codes to be used. The search for optimal codes has involved much work and continues. Hill We can be certain, however, that encoding and decoding processes must be deterministic so that the original message can be recovered exactly. These processes must therefore be implemented by deterministic computers which we model with Turing machines. Davis

We are thus led to introduce a new understanding of continuity, logical continuity, based on the mathematical notions of proof and computation. The archetype of logical continuity, corresponding to a continuous function in analysis, is a proof: a logically watertight connection between some hypothesis (say Euclid's axioms) and some conclusion (eg Pythagoras' theorem) that can be executed by a suitably programmed deterministic computer van Heijenoort. A halting Turing machine is a logical continuum, moving deterministically from an initial state to a final state.

From a practical point of view, logical continuity takes precedence over classical continuity. So our practical mathematical understanding of continuity is based on logical processes (like the epsilon-delta argument) that assume that the points in a continuous set line are so crowded together that we can always find another point between any two. Hille

Mathematical continuity by proximity motivated Cantor’s search for the cardinal of the continuum Dauben. Using logical continuity, Cohen has shown that Cantor’s approach, using theory based on sets of discrete elements, can say nothing about classical mathematical continuity. Cohen Cohen’s logically continuous argument shows that Cantor’s continuum hypothesis is independent of set theory.

5. Communication and quantum mechanics

The mathematical formalism of quantum mechanics assumes that the state space of the physical Universe can be represented by state vectors in complex Hilbert space of finite or infinite dimension. The joint state of two communicating quantum systems is represented by vectors in the tensor product space of the Hilbert spaces of the constituent systems.

The continuous evolution of state vectors in an isolated quantum system is described by unitary operators on their Hilbert space governed by Schroedinger’s equation. Since such a system is isolated, however, this continuous evolution is not directly observed but is inferred from the observed success of its consequences.

Mathematically this evolution is deterministic and reversible so that we may think of it as a process of encoding the same message in different bases. The Schroedinger equation applies equally at all energies and all levels of complexity of state vectors. The only truly isolated system is the Universe as a whole, represented in its simplest state by the initial singularity Hawking & Ellis.

The continuous evolution of an isolated quantum system is interrupted by an observation or measurement. When we observe a system, we do not see the whole continuous system, but only one or other of the basis states (eigenvectors) of the operator we use to observe the system. The mathematical formalism of quantum mechanics cannot predict which eigenvector we will observe, only the relative frequencies of the observed eigenstates.

Zurek has shown that this restriction on the completeness of observation is necessary if we are to obtain information from a quantum system. This suggests that the quantization of observation and the requirements of mathematical communication theory are consistent with one another. 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.

Further, the statistical properties of a quantum observations are identical to the statistical properties of a communication source. Like the probability of emission of the various letters of a source, the probabilities of observing various eigenstates of a quantum system are normalized to 1. This constraint is established in quantum theory by the unitarity of the evolution and observation operators. This leads us to think of the eigenstates of a quantum observation as the letters of alphabet of a communication source.

From an abstract point of view there is but one Hilbert space of each dimensionality and there is no preferred set of orthonormal basis states. The transformation approach to quantum mechanics pioneered by Dirac shows how one basis may be converted into another by unitary operators which preserve orthonormality Dirac.

We may see the communication theoretic equivalent of quantum mechanical transformations as the computational transformation of messages between different encodings using different alphabets.

6. Coding delay and special relativity

From a space-time point of view, isolated quantum systems are essentially one dimensional, parametrized by energy, frequency or time. Zee Nor do the systems described by quantum mechanics have any memory. The outcome of an observation on a quantum system depends only on the immediate state of the system and the state of the observer. Feynman

A further indication that it is plausible to consider the Universe as a communication network is provided by the structure of space-time revealed in the special theory of relativity. The events at rest in any local inertial frame are time ordered, and we believe that an event can only influence events that follow rather than precede it in time. In physics, the velocity of light is taken as the fixed and finite maximum velocity for the transmission of information from one point in space-time to another.

One consequence of the finite maximum velocity of communication is that events in an intertial frame moving relative to oneself look different. Assuming that the laws of physics are the same in every inertial frame, we may derive the Lorentz transformation which brings the distorted appearance of moving systems back to the normal appearance of things in our own rest frame.

There is nothing special about the velocity of light in this derivation. The same structures are found when we substitute the speed of the postal service, for instance, into the causal ordering of human events.

The algorithms used to defeat error in communication induce delay. In order to encode a message of n symbols into a transmissable signal function, the encoding machine must wait for the source to emit n symbols, even if the computation required for encoding is instantaneous. Since computation itself involves error free communication within the computer, we can expect it too to add delay.

We are therefore led to believe that communication delay and quantization are connected. Further, since the least delay corresponds to the maximum velocity, we propose that the algorithm used to encode information into signals that travel at the velocity of light is the fastest algorithm in the Universe.

7. ‘Spooky action at a distance’

Field theories are attempts to avoid the assumption of action at a distance where distance is understood in the ordinary spatial sense or in the relativistic sense of ‘spacelike distance’. The more powerful notion of logical continuity is not specifically related to geometric continuity. Being networked creatures we intuitively apply logical continuity in our conversations. The spatial positioning of people conversing is generally irrelevant. On the other hand to make sense, the elements of the conversation must be time ordered from the point of view of each person, which implies that there can be no conversation across a spaceline separation.

The Einstein, Podolsky Rosen thought experiment, since frequently realized, proposed that entangled particles could act upon one another at a distance even though their separation was spacelike, requiring something greater than the velocity of light to account for their correlation (if it is due to communication). Einstein, Podolsky and Rosen, Groblacher

We assume that the velocity of light is finite rather than infinite because of the delay in the transmission of a photon from point to point due to error preventative encoding. Conversely, we might speculate that communications that cannot go wrong, that is communications that effectively carry no information, might require no encoding and therefore travel at infinite velocity. The observed correlations between entangled photons have been found to propagate at many times c, and the measurements are compatible with infinite velocity, ie instantaneous transmission, over many kilometres Salart.

In practical communication networks a message originating with user A is passed down through the software layers in A’s computer to the physical layer which carries it to B’s machine. It is then passed up through B’s software layers until reaches a form that B can read. By analogy, communication between one system in the Universe must pass down to the ultimate physical layer (which we might identify with the structureless initial singularity) then up again to the peer system receiving the message.

It may be that the simplicity (high symmetry) in the lower layers of this network make encoding for error prevention unnecessary, so that instantaneous communication is possible.

8. Local gauge invariance in a network

The intersection of special relativity and quantum mechanics is quantum field theory Peskin. This intersection yields a strong set of constraints on the Universe that limit its degrees of freedom and are verified experimentally.

A fundamental symmetry of quantum field theories is local gauge invariance. From the point of view of algorithmic information theory a field theory is a system of algorithms (expressed in mathematical and computational notation) whose domain is spacetime. These algorithms transform a measured initial situation in the Universe to a measured final situation some timelike interval away. The grail is an algorithm that performs this operation without error, like the statement ‘in a closed system, energy before transformation equals energy after.

Local gauge invariance tells us that there may be a set of degreess of freedom associated with every point in space-time whose values can be set independently of their values at neighbouring points. Local gauge theories have nothing to say about these values, and so are consistent with all of them.

This feature of these theories is exactly what we expect in a compliant communication network, that is one indifferent to the actual content of the messages transmitted. This same property is manifest in the mathematical theory of communication which is indifferent to actual messages and their meaning, and is interested only in their statistical structure of their messages.

9. A transfinite symmetric network

Local gauge invariance enables us to construct a transfinite network which serves as a configuration space for the Universe. We begin with the Cantor Universe Mendelson. Cantor proved Cantor’s theorem: given a set of any cardinal n, there exists a set with a greater cardinal Cantor. Using this theorem, Cantor constructed the formal Cantor Universe by transfinite recursion. The set N of natural numbers is countably infinite. Cantor used the symbol 0 to represent the cardinal of N. The power set of N, P(N) has the next greatest cardinal ℵ1, and so on without end.

The cardinal (or power) of a set is independent of its ordering. The space chosen here, the symmetric universe (think symmetric group), is constructed by replacing the axiom of the power set in the proof of Cantor's theorem with an 'axiom of permutation' which encapsulates the properties of order and permutation Higman. The resulting structure is capable of representing any group, since the permutation group of any finite cardinality contains all possible groups of that power. We assume, using Cantors principle of finitism, that this property extends into the transfinite domain Hallett.

It is a peculiarity of transfinite cardinal arithmetic that 2aleph(n) = aleph(n)aleph(n) = aleph(n)! = aleph (n +1) so that the cardinals of peer layers in the Cantor Universe and the symmetric universe are equivalent.

We can understand a permutation as a set of n memories in which n different data symbols may be housed to give n! different arrangements. Both the data symbols and the memories can be named by the natural numbers 1 . . . n . . . .

We now imagine a locally gauge invariant field theory whose domain is the natural numbers. Local gauge invariance enables us to arbitrarily assign a natural number to each point in the domain. If each of these values is unique, we see that local gauge invariance is indifferent to the permutation of local values. We suppose that this idea can be extended to the transfinite domains of the Cantor Universe.

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 symmetric universe into the symmetric 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 symmetric universe 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.

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.

Incidentally this cyclic nature of computation may explain the ubiquity of complex (periodic) numbers and functions in quantum mechanics.

10. Formalism and the physical world

The formal structure of the symmetric network gives us a ‘mathematician’s eye’ view of the Universe. The symbols used in formal theories are assumed to be eternal (immobile) and distinct (orthogonal). From this point of view the Cantor Universe and the symmetric network exist formally in complete detail and formal interactions are instantaneous. On the other hand, the physical Universe is mobile and often uncertainly defined. How do we fit the formal model to the physical Universe?

We begin very close to the initial singularity with a state represented by a vector in a one dimensional Hilbert space whose evolution is governed by Schroedinger’s equation. Let this system interact with or observe itelf. We represent the joint state created by this communication in a two dimensional Hilbert space. Moving back the other way, we imagine the initial singularity to be a point (a vector in a zero dimensional Hilbert space) with no dynamics and completely isolated (by the definition of the Universe). This point satisfies the definition of a formal symbol.

Operations in the symmetric network, like those in everyday computer networks, are represented as strings of logical operations (a program). The structure of the symmetric network is such that every point in it is unique and every operation changes the whole Universe. In the mathematician’s eye view we see a kinematic sequence of representations of the Universe, each differing from its predecessor by intervening operations.

In this picture mathematics explains its own existence through ‘fixed point’ theorems Casti. In general, there is a fixed point in any mapping which maps a set onto itself. Since (by assumption) there is nothing outside the Universe, all its mappings are onto itself and the resulting invariant fixed points may be mapped to mathematical symbols. This makes science possible, since only immobile features of the dynamic Universe can be usefully represented by the fixed formal texts of the scientific literature.

11 Creation

Maxwell’s Demon embodied the power engineer’s hope for perpetual motion. Although we see perpetual motion in the reversible systems described by quantum mechanics, Szilard killed this hope for systems involving irreversible observation Szilard. Maxwell’s demon can do no better than a reversible heat (Carnot) engine, conserving entropy but never decreasing it.

Here we introduce Shannon’s Demon, whose task to increase entropy. Given the bad press that entropy has had over the last century, the term Demon seems appropriate to the contemporary zeitgeist.

Let us define creation as the production of new information and entropy. The fundamental axiom of information theory tells us that the information carried by a point is equal to the entropy of the space in which the point resides, so that information and entropy are created ‘simultaneously’. Defining a theory simultaneously defines a space of possible theories.

We communicate the dynamics of the world through text or formalism which, because of its stability, can be transmitted as a message. The symmetric universe is a space in which we wish to make a point about the generation of entropy, to show how the simple initial singularity differentiates itself into the complex Universe we observe today.

Let us guess that formally the same force (‘Cantor force’) makes the Universe grow as makes the transfinite numbers grow. In other words, avoidance of contradiction is the source of motion, since Cantor’s proof is non-constructive. The Universe becomes more complex because it can only do otherwise by contradicting itself. We now seek to explain the mechanism of this complexification using the theory of communication.

We begin with ‘Cantor symmetry’ that is symmetry with repsect to complexity. Cantor, explaining the genesis of transfinite numbers, wrote: . . . the transfinite cardinal numbers can be arranged according to their magnitude, and, in this order, form, like the finite numbers, a “well ordered aggregate” in an extended sense of the words. Out of 0 proceeds by a definite law, the next greater cardinal number1, and out of this by the same law the next greater, 2 and so on . . . without end Cantor, ibid. page 109.

This definite law (here represented by permutation) is indifferent to the size of the number upon which is operates to create its successor. Let us call the existence of this definite law Cantor symmetry or symmetry with respect to complexity.

A layered computer network shares this symmetry, since propositional calculus is to be found in operation throughout the system, insofar as it is observable. From the formal network point of view, complexity is irrelevant, as the theory of communication is indifferent to the entropy of the sources it describes.

Quantum mechanics also enjoys this symmetry. The formalism of quantum mechanics is essentially the same regardless of the dimensionality of the Hilbert space being used to model the system studied Feynman, ibid. In theory we can construct ‘the wave function of the Universe’ Everett III. This project may run into the sort of contradictions associated with Russell’s set of all sets. However, we anticipate no such problem constructing the state vectors of subsystems (including ourselves) of the Universe. Such state vectors can be represented in the tensor product space of all the Hilbert spaces of the constituent particles.

The defining property of each transfinite number is that it is greater than its predecessor. This suggests that transfinity is relative, and reduces our problem to the question: how does one become two, where 1 stands for n and 2 stands for >n.

We may see this as a question of resolution. Viewed from a distance, a pair may look like a unity. As we come closer, however, we resolve the individuals. In quantum mechanics that what we see depends on how we look. From this point of view, the resolution of a pair and the creation of a pair may be seen as the same operation analogous to the creation of information by the creation of the space in which the informative point or message lies.

Exploiting Cantor symmetry, we may obtain some clues to the solution of this problem from our own intellectual operations. As Misner, Thorne and Wheeler note

‘all laws and theories of physics . . . have this deep and subtle character, that they both define the concepts they use . . . and make statements about these concepts. . . . Any forward step in human knowledge is truly creative in this sense: that theory, concepts, law and method of measurement -- forever inseparable--are born into the world in union Misner.

Clear and distinct ideas emerge only slowly from the mists of ignorance. The structure of the symmetric universe is such that if we have we have resolution such that 0 = 2, then 1 = 4, 2 = 16, 3 = 2 exp 16 = 65536 and so on. The cardinals of the higher layers of the network give these layers the computing power to correct error in the lower layers, so bookstrapping their own complexity.

12. Annihilation

Maxwell’s demon fails because observation is irreversible, a fact common to both classical statistical mechanics and quantum mechanics. Special relativity opened our eyes to the relationship E = mc2 which we understand to mean that pure energy can materialise into particles and particles annihilate into pure energy.

There is perpetual motion in the reversible world described by quantum mechanics, but not in the irreversible world of quantum field theory, where particles are created and annihilated Zee, ibid.

Irreversibility is consistent with the structure of the symmetric universe. Although we speak about it in abstractions which are superpositions of many possibilities, the real world is concrete. Every memory is in a fixed state and they are all unique, so that there are as many states as memory locations. This explains the utility of permutation as a formal framework to describe the dynamics of the Universe. This situation is consistent with the quantum no-cloning theorem Wooters.

From a concrete point of view therefore, we can say that every memory is always in some state (since this is the definition of state) and that when we change the state of a memory we erase the old state while writing the new, an irreversible step.

Since it is the function of communication to change the states of memory, we find that every communication is associated with an annihilation, making communications irreversible. The way back to the starting point has been erased.

13. Determinism and indeterminism

Although Shannon’s theorems tell us that it is possible to send messages error free over a noisy channel, they do not provide explicit methods. Many methods of encoding and data compression have since been developed, some implemented in analogue circuits like FM radio, but most now depend on digital computation.

Turing devised his machine to show that there exist incomputable functions, thus answering Hilbert’s decision problem. It follows from Turing’s result that it may not be possible to compute all possible encodings.

We may imagine a communication system as a set of computable functions that can be strung together (as in an ordinary computer) to transform the input of a channel to its output. From this point of view the terms ‘computer’ and ‘channel’ mean the same. Turing machines are deterministic, so we might expect the behaviour of an isolated channel to be rather like the deterministic evolution of the wave function that quantum mechanics attributes to an isolated particle.

This determinism is broken, however, when processes communicate. Turing envisaged this situation when he described the oracle- or o-machine (ref?). An oracle machine, like any real computer, proceeds with a computation until it can go no further, at which point it halts and waits for input from an outside source or oracle. In practical communication networks, processing may also be interrupted by a message which claims priority over the current process.

A message entering a network using an accepted protocol changes the state of the machine receiving it. Given communication delay (which may effectively cause space-like separation of machines) it is impossible to predict when a machine will be interrupted and hence what the effect of the interrupt may be. Uncertainty (ie unpredictability) thus enters a network, even if all the functions driving the network are computable.

This opens the way to understanding the probabilistic nature of quantum mechanics if we interpret a quantum system as a network. In this way we replace a deterministic continuous formalism with a probabilistic interpretation with a set of deterministic machines whose communication with one another produces a set of events whose occurrences are unpredictable.

The events themselves, however, fall into equivalence classes which are determined by the communication protocols in use. The number of possible protocols is restricted to the number of possible Turing machines, that is to a countable infinity. If we consider quantum eigenfunctions to be computable, this suggests that the number of distinct eigenfunctions (ie observables) in the Universe is itself countably infinite.

Feynman was one of the first to realize that the physical processes described by quantum mechanics can be conceived as computations Feynman. Since that time, the field of quantum information and quantum computation has grown considerably and experimentalists have been able to implement some simple algorithms quantum mechanically.

14. The distinction between past and future

Until a computer has halted, its state is uncertain. Once it is halted, its state becomes definite. The same relationship exists between past and future. The past is definite, and not subject to any uncertainty principle. The future, however is uncertain, to at least the limits specified by quantum mechanics. The network model therefore suggests that the boundary between past and future can be modelled by the boundary between halted and not-halted computations.

15. Quantum computation

Many hope that quantum computers will turn out to be more powerful that Turing machines for two reasons. First, in quantum theory an operator like the Hamiltonian in Schroedinger’s equation acts on all the elements of a superposition simultaneously. This gives hope for the execution massively parallel calculations in one operation Nielsen.

Second, because a continuous variable is held to be able to encode an infinite amount of information. Since quantum amplitudes are represented by continuous functions, it is hoped that quantum computers will effectively complete operations on ‘words’ of infinite length in one operation.

To the contrary, we have the approach taken by algorithmic information theory, which measures the total information in a transformation by the length of the program required to implement the transformation Chaitin. From this point of view, the equation x = y defined on the real line does not contain an infinite amount of information, but only the few bits represented by the symbolic expression ‘x = y ’. This alerts us to the fact that the entropy we assign to a set depends on how we decide to count its elements. From a computational point of view, the algorithmic measure seems most appropriate.

15. Evolution

From a logical point of view, we are not so much concerned with the size as with the algorithmic complexity of the Universe. Although there are only a countably infinite number of different Turing machines (computable functions), the symmetric network described above is transfinitely larger, so that we may see computation as a limiting resource in the Universe. This lays the foundation for an evolutionary scheme: many possibilities, confronting limited resources select for those processes that use the resource most efficiently to secure their own survival and replication.

This process of selection may be reflected in the usefulness of the principle of least (or extremal) action in searching for candidate models of the Universe

17. Conclusion: ‘The unreasonable effectiveness of mathematics’

Eugene Wigner and many others have been surprised by the ‘unreasonable effectiveness of mathematics’ in the physical sciences Wigner. If these is any truth in the picture painted here, this fact may have a simple explanation. Mathematics is a consistent symbolic system. The stationary points (particles or messages) in the observed Universe also form a symbolic system whose consistency is guaranteed by the dynamic processes of whose limits they are. Given that the symmetric network spans the whole space of consistent symbolic systems, it may not be suprising to find that mathematics is wonderfully effective as a descriptor of the world, as Galileo proposed Galilei.

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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!)

Aristotle, and (translated by P H Wickstead and F M Cornford), Physics books I-IV, Harvard University Press, William Heinemann 1980 Introduction: 'The title "Physics" is misleading. .. "Lectures on Nature" the alternative title found in editions of the Greek text, is more enlightening. ... The realm of Nature, for Aristotle, includes all things that move and change ... . Thus the ultimate "matter" which, according to Aristotle, underlies all the elementary substances must be studied, in its changes at least, by the Natural Philosopher. And so must the eternal heavenly spheres of the Aristotelean philosophy, insofar as they themselves move of are the cause of motion in the sublunary world.' 
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Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 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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Casti, John L, Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, John Wiley and Sons 1996 Preface: '[this book] is intended to tell the general reader about mathematics by showcasing five of the finest achievements of the mathematician's art in this [20th] century.' p ix. Treats the Minimax theorem (game theory), the Brouwer Fixed-Point theorem (topology), Morse's theorem (singularity theory), the Halting theorem (theory of computation) and the Simplex method (optimisation theory). 
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Chaitin, Gregory J, Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory, World Scientific 1987 Jacket: 'Algorithmic information theory is a branch of computational complexity theory concerned with the size of computer programs rather than with their running time. ... The theory combines features of probability theory, information theory, statistical mechanics and thermodynamics, and recursive function or computability theory. ... [A] major application of algorithmic information theory has been the dramatic new light it throws on Goedel's famous incompleteness theorem and on the limitations of the axiomatic method. ...' 
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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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Dauben, Joseph Warren, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press 1990 Jacket: 'One of the greatest revolutions in mathematics occurred when Georg Cantor (1843-1918) promulgated his theory of transfinite sets. ... Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradox in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.' 
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Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. . . . The existence of absolutely unsolvable problems and the Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.' 
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Dirac, P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)  
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Everett III, Hugh, and Bryce S Dewitt, Neill Graham (editors), The Many Worlds Interpretation of Quantum Mechanics, Princeton University Press 1973 Jacket: 'A novel interpretation of quantum mechanics, first proposed in brief form by Hugh Everett in 1957, forms the nucleus around which this book has developed. The volume contains Dr Everett's short paper from 1957, "'Relativge State' formulation of quantum mechanics" and a far longer exposition of his interpretation entitled "The Theory of the Universal Wave Function" never before published. In addition other papers by Wheeler, DeWitt, Graham, Cooper and van Vechten provide further discussion of the same theme. Together they constitute virtually the entire world output of scholarly commentary on the Everett interpretation.' 
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Feynman, Richard P, and Robert B Leighton, Matthew Sands, The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Addison Wesley 1970 Foreword: 'This set of lectures tries to elucidate from the beginning those features of quantum mechanics which are the most basic and the most general. ... In each instance the ideas are introduced together with a detailed discussion of some specific examples - to try to make the physical ideas as real as possible.' Matthew Sands 
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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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Galilei, Galileo, and Stillman Drake (Translator), Albert Einstein (Foreword), J L Heilbron (Introduction), Dialogues Concerning the Two Chief World Systems, Modern Library 2001 Jacket: 'Galileo's Dialogue Concerning the Two Chief World Systems' published in Florence in 1632, was the proximate cause of his being brought to trial before the Inquisition. Using the dialogue form, a genre common in classical philosophical works, Galileo masterfully demonstrates the truth of the Copernical system over the Ptolemaic one, proving for the first time that the earth revolves around the sun.' 
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Hallett, Michael, Cantorian set theory and limitation of size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson. 
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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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Higman, Bryan, Applied Group-Theoretic and Matrix Methods, Dover Publications Jacket: '... This work, a comprehensive, thoroughly reliable exposition of the basic ideas of group theory (realized through matrices) and its applications to various areas of physics and chemistry, systematically covers this important ground for the first time. ... Although [it] deals basiclaly with advanced level material, the unusually clear exposition provides much valuable insight and fruitful suggestion for student and specialist alike. Chemists, physicists, mathematicians, and others who would like an idea of the applications and methods of group and matrix theory in the physical sciences will profit greatly from this book. ...'back
Hill, Raymond, A First Course in Coding Theory, Oxford University Press, USA 1990 Amazon Editorial Reviews Book Description: 'Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study. 
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Hille, Einar , Analytic Function Theory, Volume 1 , Chelsea 1973 Foreword: 'This book represents an effort to integrate the theory of analytic functions with modern analysis as a whole, in particular to present it as a branch of functional analysis, to which it gives concrete illustrations, problems and motivation.  
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Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.' 
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Mendelson, Elliott, Introduction to Mathematical Logic, van Nostrand 1987 Preface: '... a compact introduction to some of the principal topics of mathematical logic. . . . In the belief that beginners should be exposed to the most natural and easiest proofs, free swinging set-theoretical methods have been used."  
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Misner, Charles W, and Kip S Thorne, John Archibald Wheeler, Gravitation, Freeman 1973 Jacket: 'Einstein's description of gravitation as curvature of spacetime led directly to that greatest of all predictions of his theory, that the universe itself is dynamic. Physics still has far to go to come to terms with this amazing fact and what it means for man and his relation to the universe. John Archibald Wheeler. . . . this is a book on Einstein's theory of gravity. . . . ' 
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Nielsen, Michael A, and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2000 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002. 
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Peskin, Michael E, and Dan V Schroeder, An Introduction to Quantum Field Theory, Westview Press 1995 Amazon Product Description 'This book is a clear and comprehensive introduction to quantum field theory, one that develops the subject systematically from its beginnings. The book builds on calculation techniques toward an explanation of the physics of renormalization.'  
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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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van Heijenoort, Jean, From Frege to Goedel: A Source Book in Mathematical Logic 1879 - 1931. , iUniverse.com 1999 Amazon book description: 'Collected here in one volume are some thirty-six high quality translations into English of the most important foreign-language works in mathematical logic, as well as articles and letters by Whitehead, Russell, Norbert Weiner and Post…This book is, in effect, the record of an important chapter in the history of thought. No serious student of logic or foundations of mathematics will want to be without it.' 
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Zee, Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2003 Amazon book description: 'An esteemed researcher and acclaimed popular author takes up the challenge of providing a clear, relatively brief, and fully up-to-date introduction to one of the most vital but notoriously difficult subjects in theoretical physics. A quantum field theory text for the twenty-first century, this book makes the essential tool of modern theoretical physics available to any student who has completed a course on quantum mechanics and is eager to go on. Quantum field theory was invented to deal simultaneously with special relativity and quantum mechanics, the two greatest discoveries of early twentieth-century physics, but it has become increasingly important to many areas of physics. These days, physicists turn to quantum field theory to describe a multitude of phenomena. Stressing critical ideas and insights, Zee uses numerous examples to lead students to a true conceptual understanding of quantum field theory--what it means and what it can do. He covers an unusually diverse range of topics, including various contemporary developments,while guiding readers through thoughtfully designed problems. In contrast to previous texts, Zee incorporates gravity from the outset and discusses the innovative use of quantum field theory in modern condensed matter theory. Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used.  
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Papers
Groblacher, Simon A, Tomasz Paterek, Rainer Kaltenbaek, Caslav Brukner, Marek Zukowski, Markus Aspelmeyer, & Anton Zeilinger, "An experimental test of non-local realism", Nature, 446, 7138, 19 April 2007, page 871 - 875. Abstract: 'Most working scientists hold fast to the concept of 'realism'—a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell's theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of 'spooky' actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned.'. back
Salart, Daniel, et al, "Testing the speed of 'spooky action at a distance'", Nature, 454, , 14 August 2008, page 861-864. 'Correlations are generally described by one of two mechanisms: either a first event influences a second one by sending information encoded in bosons or other physical carriers, or the correlated events have some common causes in their shared history. Quantum physics predicts an entirely different kind of cause for some correlations, named entanglement. This reveals itself in correlations that violate Bell inequalities (implying that they cannot be described by common causes) between space-like separated events (implying that they cannot be described by classical communication). Many Bell tests have been performed, and loopholes related to locality and detection have been closed in several independent experiments. It is still possible that a first event could influence a second, but the speed of this hypothetical influence (Einstein's 'spooky action at a distance') would need to be defined in some universal privileged reference frame and be greater than the speed of light. Here we put stringent experimental bounds on the speed of all such hypothetical influences. We performed a Bell test over more than 24 hours between two villages separated by 18 km and approximately east–west oriented, with the source located precisely in the middle. We continuously observed two-photon interferences well above the Bell inequality threshold. Taking advantage of the Earth's rotation, the configuration of our experiment allowed us to determine, for any hypothetically privileged frame, a lower bound for the speed of the influence. For example, if such a privileged reference frame exists and is such that the Earth's speed in this frame is less than 10-3 times that of the speed of light, then the speed of the influence would have to exceed that of light by at least four orders of magnitude.. back
Szilard, Leo, "On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings", Behavioural Science, 9, 4, October 1964, page . 'In memory of Leo Szilard ... we present an English translation of his classial paper Uber die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen which appeared inthe Zeitschrift fur Physic 1929, 53, 840-56. This is one of the earliest, if not the earliest paper, in which the relations of physical entropy to information (in the sense of modern mathematical theory of communication) were rigorously demonstrated and in which Maxwell's famous demon was successfully exorcised: a milestone in the integration of physical and cognitive concepts. ' Reprinted in Feld, Bernard T, The Collected Works of Leo Szilard: Scientific Papers, The MIT Press 1972 Amazon  back . back
Wigner, Eugene P, "The unreasonable effectiveness of mathematics in the natural sciences", Communications in Pure and Applied Mathematics, 13, 1, February 1960, page 1-14. '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 ofour physical theories.'. back
Wooters, W K, W H Zurek, "A single quantum cannot be cloned", Nature, 299, 5886, 28 October 1982, page 802 - 803. Abstract: 'If a photon of definite polarization encounters an excited atom, there is typically some nonvanishing probability that the atom will emit a second photon by stimulated emission. Such a photon is guaranteed to have the same polarization as the original photon. But is it possible by this or any other process to amplify a quantum state, that is, to produce several copies of a quantum system (the polarized photon in the present case) each having the same state as the original? If it were, the amplifying process could be used to ascertain the exact state of a quantum system: in the case of a photon, one could determine its polarization by first producing a beam of identically polarized copies and then measuring the Stokes parameters1. We show here that the linearity of quantum mechanics forbids such replication and that this conclusion holds for all quantum systems.'. back
Links
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 some 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 of a real or computable variable, computable predicates and so forth. . . . ' 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
Claude Shannon, Communication in the Presence of Noise, 'A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two “function spaces,” and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of “ideal” systems which transmit at this maximum rate are discussed. The equivalent number of binary digits per second for certain information sources is calculated.' back
Einstein, Podolsky and Rosen, Can the Quantum Mechanical Description of Physical Reality be Considered Complete??, A PDF of the classic paper. 'In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false, One is thus led to conclude that the description of reality given by the wave function is not complete.' 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
Wojciech Hubert Zurek, Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical, 'Submitted on 17 Mar 2007 (v1), last revised 18 Mar 2008 (this version, v3)) "Measurements transfer information about a system to the apparatus, and then further on -- to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide framework for the ``wavepacket collapse'', designating terminal points of quantum jumps, and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment -- the fittest quantum states that not only survive decoherence, but subvert it into carrying information about them -- into becoming a witness.' back

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