Notes
[Notebook: DB 60 Spotlights]
[Sunday 1 April 2007 - Saturday 7 April 2007]
[page 159]
Sunday 1 April 2007
What this adds up to is that we consider Einstein's field equations as describing the protocol of the universal network. So our first step is to map a differentiable manifold to the transfinite network. So we map points in the manifold to communications (events) in the network. Next we interpret the operations of differentiation and integration as stepping through peer layers in the network. Differentiation exhibits detail and so increases cardinality (complexity, entropy). Integration obscures detail by adding it all together and so decreases entropy. The messages transmitted through the network are encoded in the connection coefficients and the rate of transmission in 'acts per second' maps to energy which in turn maps to the curvature of space, the magnitude of the connections, their strength, their 'charge'. Something like this, keep rattling all these pieces around until they fall into place, just as we see happening in the molecular biology of life. It works because the 'correct place' has less potential energy that all the other places and so right places tend to bind and release their spar energy in emitted particles.
With this in mind, the trick is to fit the actual Einstein equations to a model of the Universe, ie
[page 160]
'theory of everything' or theology. So first have to understand the equations gmn. These equations establish symmetries in the network that enable any message that quantum mechanics to dream up to be transmitted. Here is the physical layer of the Universe.
Space of n dimensions = space of n degrees of freedom among which we seek 'couplings' where the value of one is some function of the value of others.
Observer = I
Observed = YOU
Observation = I and YOU = {I, YOU}
Either the points of the manifold themselves or the coordinates can play the role of I and YOU.
I is the observer, YOU is the observed, but to observe is to be observed and vice versa. An observation is an entanglement of I and YOU which cannot be understood in terms of I or you alone.
Consider as a manifold a 'dust', every one of whose elements is parametrized by a 'proper time' which initially is known only to the grain of dust in question. We say that di can only check its clock by communicating with some other grain, dj and comparing proper times. In this way we can set up a network of calibrations, but the whole system is constrained by our decision to only allow the grains to communicate pairwise.
[page 161]
This means that if dj wants to compare itself with di <> j, it must do so sequentially, at different proper times. To gain an overall picture of its environment, di must have memory and the ability to transform values stored at different points in its memory into one another. In this example, di can do this if it knows how to add and subtract [units of time].
Without coordinates, ie a benchmark for a relationship, there can be know knowledge. The knower sets up he coordinates that yield the most useful information, we look for what we need to see to survive.
So all experience is messages in fourspace, each with a certain volume defined as bandwidth by time.
Hobson page 26: A manifold is differentiable if it is possible to define a scalar field at each point in the manifold that can be differentiated everywhere.' Hobson
In the transfinite network viewed as a manifold, this scalar field might be entropy whose gradients steer information one way or another.
The 'coordinates' of a message (event) give us its position in spacetime and derivatives of these coordinates with respect to the proper time of the event tell us how it is moving.
page 27: 'In general a set of coordinate patches that covers the whole manifold is called an atlas'.
[page 162]
Since every observer has its own coordinate system (we make this a definition) the atlas of the Universe must contain the coordinate systems belonging to every observer in the Universe. Of all possible observers, only are a subset are in existence in a given subset of the Universe.
Peer groups are subspaces of the transfinite manifold.
Hobson 28 General covariance tells us that the choice of coordinates is arbitrary, and so the structure of observations may be arbitrary, at least if we overlook the need to survive.
page 28: Coordinate transformation relates points in a manifold, ie shows us how to connect the points of view of different observers.
page 28: 'The important idea s not the 'labels' but the points themselves.' In the transfinite network each point is a label and the whole business becomes a set of formal transformations between arbitrary points of view.
From a communication point of view, a coordinate transformation is a channel which conveys the output of a source to the input of a sink. ['source' and 'sink' denote the points where potential energy becomes kinetic, and so observable, if only by each other]
page 29: 'We will assume that the functions involved [in a coordinate transformation] are single valued, continuous, and differentiable over the ranges of their arguments.'
[page 163]
Such transformations may be said to conserve entropy by being single valued.
The physical layer of the Universe is where energy and entropy meet.
We move in an ambience of information, that is realized structure which guides motion but does not determine it, since without degrees of freedom their cannot be motion.
Hobson page 29: The elements of the transformation matrix are partial derivatives of the new coordinates with respect to the old functions of the coordinates.
If, in human psychology, we understand an act of insight as a coordinate transformation in the mind (epitomized by Aquinas' commentaries on Aristotle) we may think of the transformation matrix as a set of logical functions that transform one representation of a 'truth' (sentence) into another. My whole task is to transform all human knowledge from the basis that the Universe is not-God to the basis that it is God.
Monday 2 April 2007
In General Relativity the transformations re something deterministic which can be performed (at least approximately) by a Turing machine. At the lowest level of resolution, the numerical values in the transformation matrix come from the set {0, 1}, ie they are one of the two binary digits.
[page 164]
My sensory system transform a swarm of photons entering my eyes into the statement 'the cup is on the table'
Hobson page 30: Einstein summation convention = integration convention.
Infinitesimal distance measurement in the 'reference mollusc'
page 31: GEOMETRY == DISTANCE
The 'variety' of general covariance is reduced by symmetry.
If everything is represented by binary strings and operations that preserve the lengths of these strings, we have a little Universe whose geometry may be based on the 'Hamming distance' which relates to the probability of confusing two strings in this Universe. When Hamming distance is zero, the two strings are identical, and the distance is a maximum between strings that differ in every position. We can extend this idea to measuring the distance between permutations.
Infinitesimal difference = 1 bit. Distance depends on the 'significance' of the differing bit. In General relativity al 'bits' have the same significance. In the transfinite network distance depends
[page 165]
on the 'meaning' to the 'user' of the differing symbols.
This idea is captured by the metric tensor in Riemann's geometry, where ds2 = gab(x) dxadxb. where the elements of gab give 'weight' to various elements of the vectors xa. gab is symmetric gab = gba, which follows from the commutativity of ordinary multiplication. As a result the metric has 10 independent function, 4 on the diagonal and 6 off.
Hobson page 33: 'Since there are N arbitrary coordinate transformations there are really only 1/2 n(n-1) [= 6] independent degrees of freedom associated with the gab(x).
Continuous functions are a subset of all functions. If we see all functions as permutations, we may say continuous functions are those that swap neighbouring (local) values.
We note here (notices somewhere else about 20 ya) that a Turing machine is a 4D object, ie it has 4 degrees of freedom.
We might expect to find hat the more physical constraints on the structure of the Universe will have lower cardinality, which might be represented by the spin of the particles exchanged. By a quirk of fate photons (spin 1) have two rather than 3 states (because of their velocity), the same cardinal as the 2 state electron, spin 1/2.
[page 166]
All that constrains in the number of degrees of freedom. Systems will then develop the maximum variety available given the [?].
Invariance with respect to complexity happen 'within' cardinals (like the natural numbers) as well as between the transfinite cardinals, the alephs which give is a scale of degrees of freedom against which to measure the Universe.
We suppose that the actual functions implemented in the Universe (out of all possible function) are selected by some evolutionary process. Can this explain why the fixed points in quantum mechanics are the eigenvalues of the operators that implement the transformations of quantum state vectors. We might guess that eigenfunctions are valuable as basis vectors because problems formulated in this natural basis are easier to solve and so confer some sort of survival value (and over representation in the population) of sexier measurements, algorithms and actuator.
Looking at this, we may guess that algorithms corresponding to similar mathematical expressions are expressed in the high frequency low complexity world of physics.
The mathematical models of physics mimic the behaviour of the Universe. Why is this so? Because the Universe itself is the execution of a mathematical model which has the incompleteness, incomputability and potential paradox that mathematics itself predicts to be a feature of all formal systems. The formal system of the Universe is well tuned because it is big, old and fast, reasonable
[page 167]
mimicked in many areas by Newtonian mechanics which allows for instantaneous communication and computation, without the delays inherent in relativity and quantum mechanics. In physics, these delays only become important under fairly extreme conditions. But in real life, where the local 'velocity of light' is very much slower, we know that time (the measure of delay) is often of the essence in survival situations.
Every complex process takes time, the measure of before and after (Aristotle). Proper ordering of events (Proper time) Einstein.
Hobson page 34: The surface of a cylinder is not [intrinsically] curved.
ABSTRACT = ZOOM OUT
CONCRETE = ZOOM IN
Line element = message
The symbolism leads the intuition when we come to deal with 4-space.
A geodesic is the path of a 'free' (inertial) message through the network.
page 38: 'The lengths of curves follow immediately from the line element.'
page 50: The field equations of general relativity do not constrain the global topology of the spacetime manifold.' which is an entanglement of loops of communication at all
[page 168]
scales.
. . .
Coordinate independent = symmetrical with resect to coordinates, ie blind to them. I do not care about how they are looking at me. Feynman: What do you care about what other people think.
. . .
Tuesday 3 April 2007
Taking advantage of their common 4-dimensionality can we map Minkowski space to the Turing machine?
[page 169]
We may speak of the 'components' of a Turing machine.
Hobson page 60: Basis vectors are an alphabet.
page 62: 'a null vector is a non-zero vector which is orthogonal to itself'.
'In order to define the derivative of a vector field . . . one must compare vectors at different points, albeit in the limit where the distance between the points tends to zero.'
In the discrete model, a vector represents the state of a Turing machine and we can calculate the derivative of such a vector by comparing adjacent states of the machine, ie states one operation apart.
Practical advice: we are inclined to see meaning (often malicious) where the is none and this can lead to trouble.The modern version of the Buddhist 'pure mind' is 'evidence based mind'. How do we extrapolate this idea to physics? We say that more complex physical structures (like an atom) are evidence based insofar as they are built on the properties of their constituents, electrons, protons, neutrons, etc. By respecting the properties of its constituents (which constitute an alphabet) the coalescence of these parts opens up a whole new space of permutations of this alphabet.
This is our notion of democracy and justice, that all the properties of every individual be honoured in the construction of our pairwise relationships which
[page 170]
form the foundation for the construction of our society,..
In communities with immigration we can conduct epidemiological studies to discover which traits of human behaviour are learned from one's human environment an which are 'alphabetic' traits, properties of 'human nature' in the biological sense.
We see the transfinite network as a space for sociology as well as for physics. We use features of the transfinite network to construct a world which may be compared wit the world we inhabit (which includes these ideas).
Hobson page 62: Connections: here's the wormhole to the network picture = network coordinates. We may look on the transfinite network and its transformations as a coordinate system against which to measure our experience.
The purpose of this article is to introduce a new paradigm for the understanding of our world. In essence, this paradigm change revolves around understanding the term measurement as used in science as communication between the scientist and whatever she s trying to understand.
Further, although the term measurement tends to connote a communication with a scientist on one end and the world on the other, our paradigm assumes that every event in the Universe is a communication. This paradigm holds regardless of the complexity of the event, which may range form the emission of a photon to the lifetime of the Earth and beyond.
[page 171]
So I see myself as a communication, and my planet, and in fact the whole Universe as a vast network of communication. To be understood by analogy with computer networks.
There is some tendency to aught at 'ancient wisdom' but first we must make sure we understand it as it was understood by its writers. So we can build on some at least of Aristotle Aquinas and their ilk.
Hobson page 66: Gamma abc is the connection constructed from derivatives of the metric. Sixteen components of the metric, by differentiation, become 64 components of the connection.
CONNECTION = CONVERSATION
DIFFERENTIATION = COMPARISON (SUBTRACTION) = MEASUREMENT
. . .
Control = gardening: preventing or killing unwanted growth, promoting wanted growth, = farming.
In the infinitesimal realm we concern ourselves with linear effects only on the understanding that for numbers less than 1 higher powers become negligible. In mathematics, we let delta x go to zero in order to get dx. We can quantize calculus however, by letting delta x f=go not to zero but to the 'quantum of arithmetic' analogous to the quantum of action. Let us guess ha the size of such a quantum would be the inverse of
[page 172]
the cardinal of the Universe of our discourse.
We define an array b the set of addresses where the variables are stored. Every unit of measurement has a physical and a logical address. Does gravitation assign logical addresses to physical particles?
Quantum mechanics - discrete events have continuous probability die to the high cardinal of the processes of which they are part.
GOD = SOURCE
GOD == THE SOURCE
SUBTRACTION - DIFFERENTIATION
ADDITION - INTEGRATION
Adding, subtracting, differentiating, integrating what?
Units, sets, symbols, all sorts of formal constructs. That is mathematics. Our 'message symmetry' suggests that the world probably works in the same way that mathematicians work, and tends to come to the same conclusions: it makes sense to use the eigenvalues of an operator to construct a basis from which we can measure other states and superpositions of states.
Symmetry with respect to complexity puts mathematicians and atoms and every other communicating entity in the same boat.
[page 173]
We explain creation by the power of error correction that is control. These powers are formally explicated in the mathematics of cybernetics.
What structures can propagate through our 4-space? In the old days it was vibrations of the ether (string theory).
The elements of the metric are differentials, numbers representing rate of change of two vector components with respect to one another.
STRUCTURE = MEMORY = TIMELIKE LENGTH = SPACELIKE IMMOBILITY = BONDING = ONE-ONE CORRESPONDENCE.
Wednesday 4 April 2007
CONTROL - POTENTIAL. It is the potential that controls the structure of the world? Or at least half the control, see eg electronic structure of an atom.
An 'out of control' government spurns the advice of its bureaucracy, its scientists and all the others who pay close attention to the navigation of the ship of state, and announces projects costing billions because they are believed to be popular with the electorate and so buy votes for a government prepared to throw away a deep understanding of history on its lust for power.
[page 174]
Thursday 5 April 2007
Computation (calculation) (proof) means systematically replacing a given expression by a logically equivalent expression, ie 2 x 2 = 4. Logically equivalent (formally equivalent) being defined by some set of axioms.
Hobson: we break the global symmetry of etaab to get gab, which varies from point to point.
No connection = no communication = no exchange of gravitons. Conversely, once communication is started in the Universe, curvature is induced, and from the universality of gravitation this can be any communication, leading to the equation
COMMUNICATION = GRAVITATION
MESSAGE = GRAVITON
We say that the graviton has 'spin 2', ie five states |-2>, |-1>, |0>. |1>, |2> where the digits are names without any connotation of quantity.
Zee page 16: We meditate on the difference between the Hamiltonian H and the Lagrangian. H = KE + PE, L = KE - PE where KE = kinetic energy PE means potential energy. Kinetic energy is always positive, being the product of two positive numbers, m and v2. Unless we allow v to be a complex number, v2 is always positive. PE, on the other hand, breaks this symmetry and we represent this broken symmetry (or differentiation) by putting the signs + and - in front of PE. As a consequence, H is conserved, L is not.
If we think of kinetic energy as processing rate (actions per second) how do we understand potential energy? Let us say that it is the energy of form or structure.
Since de Maupertuis' time, we have believed that God made the Universe perfect and therefore efficient, taking the least possible action to achieve its ends. Yourgrau
In our vacuum, the path integral measures the numbers if single character exchanges needed to move from the permutation representing A to the permutation representing B. General covariance says we can make any permutation represent any point as long as we conserve the ? distance between points. This is where the metric constrains the logical, so that all our logic is implemented in 4- space, but 'explained' by what goes on in n-space (eg Hilbert space, mathematical space)
Mathematical space = anywhere mathematics can go = (we hope) the transfinite network. We work things out in the transfinite network and apply the results to the management of the networks we rely upon for health and happiness.
Zee page 20: Disturbing the vacuum. Zee Since the vacuum represents chaos, any disturbance to be observable must be in the direction of order. Science has no 'explanation' for chaos since it admits no explanation.
[page 176]
It is incompressible in terms of the lengths of the strings necessary to represent it (Chaitin). Chaitin Disturbance introduces order and therefore compressibility. 'Print 0 1000 times' is a much shorter string than the 1000 0's which it represents.
To disturb the vacuum we need a source of disturbance. Is this source distinct from the vacuum or is the vacuum its own source? If we want to represent the vacuum by the transfinite network, this seems quite likely, since it contains every group of every cardinality and so is potentially compressible in certain directions, ie when ever a structure can be encoded in something smaller than itself. Such a situation arises in the context of life where the specific structure of a living creature is compressed into a genotype which must be decoded within a living cell to have effect.
GENOTYPE = POTENTIAL
In general potentials constrain thing, My local gravitational potential keeps me in my chair and makes it necessary for me to exchange energy with my environment if I want to go up or down, although I can (theoretically) move sideways without effort.
Energy is conserved. So what is energy? Something that stays the same through time, parallel transported along the time axes. Here we se the Universe as a network of computational steps whose minimum size is measured by the quantum of action.
[page 177]
zee pa 21: He discretizes spacetime into a lattice with a certain (arbitrary) spacing. Our lattice is the network w, which has a natural minimum, the transmission of one message, measured by Planck's quantum of action. The points in our lattice are addressed by permutations.
So the generation of chaos endows it with a certain 'Platonic' structure of symmetries which have some sort of reproductive nature which allows them to grow at the expense of other structures and so introduce order into chaos.
The transmitter loses energy, the receiver gains it. Conservation of energy requires that the transmitter's loss is exactly matched by the receiver's gain.
Action (angular momentum) is conserved as well as energy but the two are lot linearly coupled. Each time a photon is emitted or absorbed by an atom its internal angular momentum changes by one unit, but the energies of the photon depend of the change of state to which they are coupled and may range from gamma to microwave.
Messages are the alphabet of the network.
Our hope lies in that fact that given any situation (point) aleph(x) we can find a larger system aleph)x+1) which has sufficient power to remove all errors relative to a given structure in aleph(x) (eg me). With enough peace and medical care, I can lead a pretty good life, and the have at cardinal aleph(me) is maintained by the structures in the higher cardinal aleph(me+n).
[page 178]
Curved (ie dynamic, flexible) space has enough degrees of freedom to model the large scale structure of the Universe whereas static (Cartesian) space does not.
One marks a path through a book, but there is an effective infinity of such paths available to the reader.
TEXT = POTENTIAL. All texts are dual, here the black ink and the white page. Many are of higher cardinal created by the permutation of simpler texts. The beautiful things about potentials is that they superpose. [ie all the laws of the land are simultaneously operative].
Subtraction is essential for error detection and correction, since one must subtract achieved from desired (by the potential) to get the error signal to apply to the correction algorithm.
We may identify the physical alphabet, the chemical alphabet, the biological alphabet and many graduations in between.
The eigenvalues of an operator are like graduations on a tape or the transduction capabilities of the light sensitive cells in out eyes, constraining what can be observed to the capabilities of the observer, a form of pixellation.
When physics speaks of 'some arbitrary function f' we assume that f is computable so that it can actually give results.
[page 179]
Let us assume that the transfinite network is incompressible almost everywhere. The countable number of exceptions to this incompressibility establish symmetries in the Universe that allow for memory and computation, and so the birth of stable processes, ie cyclic, recursive, 'wave'.
We might say that the transfinite network has a countable number of eigenvectors, that is possible messages at the physical layer. From these vectors as a basis, the rest of the Universe is built. The eigenvectors can be placed into correspondence with the natural numbers (and their corresponding eigenvalues).
Monastery: an adiabatic enclosure designed to fence out 'the world' so that its inmates can come closer to God.
It is very difficult to top quantum systems interacting, so if we want total control for computing purposes, we must learn to build adiabatic walls and correct for any outside influences that do get through.
Friday 6 April 2007
Saturday 7 April 2007
Nielsen re Shor code (page 434): 'This is an example of the extraordinary fact that the apparent continuum of errors that may occur on a single qubit can all be corrected by correcting only a discrete subset of these error; all other possible errors being corrected automatically by this procedure. This discretization of errors is central to the way quantum error correction works and should be regarded in contrast to classical error correction for analog
[page 180]
systems where no such discretization of errors is possible.' Nielsen
This suggests that the idea that a qubit contains an infinite amount of information is flawed (?).
