Notes
[Notebook: TURKEY]
[Sunday 27 april 2003 - Saturday 3 May 2003]
Sunday 27 april 2003
Monday 28 April 2003
[page 269]
Tuesday 29 April 2003
The mathematical models used by physics can mostly be found at the interface between ℵ0 and ℵ1. Is there any need to go beyond this, ie is there any need for transfinite dynamics in physics? The only leads we have seem to be first the composition of Hilbert spaces and second the higher orders of the Feynman diagrams for QED etc. The basic need for the transfinite network, from this point of view, is purely theological. Yet thee is hope for it, since it is the largest possible dynamical system, and so the constraints and symmetries in it are minimal and show us the way to understand the structure of god.
CORRESPONDENCE PRINCIPLE for transfinite dynamics and standard dynamics comes from the CANTOR SYMMETRY that tells us that the relationship between ℵ0 and ℵ1 is a paradigm for the relationships between aleph(n) and aleph(n+1).
[page 270]
'Fictitious' forces are produced by 'fictitious' frames of reference - ie the tendency (force) to evil is introduced by the Christian reference frame that postulates a very restricted version of human behaviour.
Lawden: 'When equations preserve their form upon transformation from one reference frame to another, they are said to be covariant with respect to such a transformation. Newton's laws of motion are covariant with respect to transformation between reference frames. Lawden.
A 'state of motion' exists in itself (ie I am a state of motion) but is measured and communicated with respect to a certain reference frame, in the human case a language eg X is angry, which can appear in many different languages as expressing the same 'emotional state'.
What is the frame of reference in communication? The relationship between the points of communication represented by a shared communication protocol (language) eg English, gravitational strong force, [cell] signalling pathways.
Special principle of relativity: all physical laws are covariant with respect to transformations between inertial frames (and if not so are not physical laws?)
NEWTON: One inertial frame does for all.
EINSTEIN 1: A set of inertial frames does for all.
EINSTEIN 2: All frames, inertial or not, are equivalent - general covariance.
The human frame of reference is society. I measure myself and others against social norms, customs, laws.
Physical properties are defined by the method of measurement [communication].
What we really want, though, is a set of symmetries to govern political dynamics. We need a transformation that (if a
[page 271]
society is humane, (at a given peer index) all asymptotically consistent societies can be transformed into one another.
So we talk about 'peer transforms'
We begin to talk about the transfinite network by considering ordinary computer networks.
Can we make economics isomorphic to physics energy = money (or money flow)
In mathematics, the = sign signified the equivalence of two strings of symbols under a certain transformation, ie when they are given certain meanings. The meaning, and so the specification of a transform comes from higher, more stable layers. This maps nicely onto the transfinite network, since the rate of change of massively parallel systems is slow compared to that in simple systems.
In one's proper time, space and time take their 'natural' pace.
The uniqueness theorem - every proper time is unique, a decomposition of action = energy.time. So in a sense w make God (the infinitely slow) the reference from which all faster processes are given meaning, which is modelled by the largest (relevant) peer index. There is no largest peer index, but there is a smallest, 0 which corresponds to a dust of unconnected blocks. We must see general covariance and gravitation as a manifestation of this divinity.
SARS: One of the most important features of democracy is the continual broadcasting of accurate information about the state of the union, so that individuals can adjust themselves to the whole. This is analogous
[page 272]
to the interaction of fields (at least infinite ones) and particles (eg EM and gravity). Here we speak of a 'fully informed peer group' ie a set of indices representing the state of individuals are summed and then the sum or normalized sum broadcast so that each individual can see where it lies on the global spectrum.
In the political sphere one would like to use a variational principle to detect the geodesic location between conservative and radical.
In general, we say, spirit provides a reference frame for matter, ie aleph(n+1) for aleph(n).
Corruption: unnecessary operations (from the system point of view) which may benefit subsystems (leakage of public energy into private tasks).
Lawden page 29: By expressing a physical law as a tensor equation, we shall guarantee its covariance with respect to a change of inertial frame.
In a conserved system we can find conserved parameters which measure the conserved features of the system,.
Manifold: set of points with smooth addressing - 4D = 4 numbers per label.
General covariance: It makes no difference to the physics what reference frame one uses.
All observables are invariant with respect to coordinates et time, energy, momentum. Since two reference frames agree on invariant, and invariant are what we observe, so coordinate systems are observationally indistinguishable.
Wednesday 30 April 2003
[page 273]
John Norton Hole Argument.
Leibniz Equivalence: If two distributions of fields are related by a smooth transformation, then they represent the same physical system [ie only discontinuities make a difference.]
We assume that the initial singularity is connected to the contemporary Universe, and like it, has by definition nothing outside it to constrain it. We therefore expect it to explore the whole territory of consistent states. The only boundary, and so the only source of structure, is consistency.
We may consider the current Universe as evolving inside the initial singularity and the spatial extension we observe to be the product of an increase in structural (potential entropy) at the expense of temporal entropy. The passage of time lays down space driven by Cantor's theorem.
Following Kronecker's dictum that God made the naturals, we give the initial singularity an entropy of ℵ0 corresponding to a count of ℵ1, since S = k log W.
Thursday 1 May 2003
Hawking and Ellis The Large Scale Structure of Spacetime. Hawking and Ellis.
We begin with gravitation because it is universal [so is quantum mechanics!] We presume that it represents the outcome of logical confinement and so is present even in the initial singularity. We presume also that gravitation acts as a superposition of two time directions, one of which leads to expansion (big bang) and the other contraction (galaxies, stars, planets, black holes etc)
Laplace 1798
Since we are inside the initial singularity, it and its
[page 274]
laws are present at every point in our world, although its energy density is now very low so that in any spacetime volume these laws are executed at a much slower rate than in the original singularity. SERIAL PARALLEL transition = transfinite transition.
Einstein took his stand in the inertial frame which is nevertheless not a closed entity but rather needs a space in which to move freely to be inertial? Is this so? Perhaps the initial singularity is inertial because there is nothing outside it to stop it being inertial! A starting point attractive to me is spin because intuitively something can spin without going anywhere, ie it is naturally 'closed'. Also we find that the measure of spin, angular momentum, has the same dimensions as Planck's constant, the quantum of action, so we consider this to be a measure of the elementary operations of the Universe. [as spin increases, degrees of freedom increase (Pauli) Such operations we consider to be represented formally by the execution of a Turing machine.
Hawking page 4: 'If the energy-momentum tensor satisfied certain positive definite conditions, [the tidal force] always has a net converging effect on non-rotating families of geodesics' But what about rotating? Does the initial singularity spin itself into complexity? What is the total spin of the Universe, ℵ0 or 0? If zero, spin adds algebraically and cancels.
5: Longest time = lowest energy (for stationary action)
Durable = consistent over time ie cyclic. Income >= outgoings
page 6: We shall mostly consider spacetimes which do not permit causality violations. In such a spacetime, given any space like surface S, there is a maximal region of spacetime (called the Cauchy development of S) which can be predicted from knowledge of data on S. A Cauchy development has a property (global hyperbolicity) which implies that if two points in it can be joined by a time-like curve, then there exists a longest such curve between the points. This curve will be a geodesic.
Does this mean that motion in time is cheaper than motion in space?
[page 275]
Do nothing gets me through time. To move [spatially] relative to my environment means I have to do something. Potential. [potential through space; potential through time = energy = rate of action = probability of something happening = frequency x probability per trial]
First law: proper time elapses no matter what. So four-momentum in the absence of space is energy.
General covariance says all consistent reference frames are equivalent, but the notion of measurement says what is measured and the reference frame must have the same cardinal number, ie the same amount of detailed structure. The existence of reference frames (languages) points, therefore to the structure of the world, and the cardinal reference frame maybe considered unique since it is concerned with entropy alone and does not deal with meaning.
[general covariance = physical events do not see their own meaning - Chaitin's principle]
MEASUREMENT = COUNT
Hawking and Ellis page 6: 'The causal structure of space-time can be used to define a boundary or edge of spacetime' [no influences come from beyond the boundary.]
page 7: In Chapter 8 we discuss the definition of spacetime singularities. This presents certain difficulties because one cannot regard the singular point as being part of the spacetime manifold M.'
Why? We prefer the singularity to be present at every point in the manifold.
[is this a measure question: the initial singularity has measure zero in the spacetime manifold?]
Conditions for singularity:
1. gravity attractive
2. enough matter to prevent anything escaping, which is the same as
saying nowhere to go, ie the singularity is in some sense
complete.
3. no causality violations? What about superposition which sees an
infinity of different (= inconsistent) structures existing at a
point.
page 11: Manifold = set of particles + smooth addressing.
Can we speak of a manifold of permutations? Hamming distance. Min distance = 0, ie distance of
[page 276]
a permutation from itself. Maximum - ℵ0, two permutations differing at every point.
page 11: Manifold is defined by a set of numbers* (coordinate system) but to be really different points must have some intrinsic ID, otherwise they would be identical! Intrinsic ID introduced 'countability' and 'mappability'
* Rn = {n-tuples}, ie independent dimensions. So we begin with distinct instances of time.
Hawking and Ellis page 15: A curve is a map of the real line into a manifold. A 'discrete curve' (eg an infinite series) as a map of the natural numbers into a manifold.
Given P0 with no structure (like god) we induce diversity by relationship, as in the Trinity.
. . .
Mathematical definitions are invariant with respect to natural language.
Transfinite dynamics exposes the spectrum of complexity inherent in the real line, as Cantor (see Jourdain). Cantor, Jourdain's introduction, pp 1 - 82. [we see the big bang as the 'complexification of the line. Time seems to be the parameter of this line, perhaps in complex space]
We make the mapping Cinfinity = Cantor Universe.
General covariance: all the pretty structures in aleph(n) are different ways of giving meaning to the elements [structures] of aleph(n-1).
Transfinite dynamics must begin with transfinite statics. The Cantor Universe is a picture of transfinite statics. We add Turing machines to give transfinite dynamics [partitioning computations into deterministic and indeterministic] We us probability theory to make some predictions about this system and explain evolution.
Axiom: any ordering is a representation of the transfinite transition. ORDERING = BONDING.
[page 277]
ORDERING = BONDING = MEANING
Although order is said to be the opposite of chaos, it is also the source, since we create a vast increase in entropy/energy, by creating ordered sets of symbols, ie in an ordered system, entropies add.
Hawking and Ellis page 15:
Covariant vector is an operator which maps any once differentiable function f at a point on a on a curve in a manifold to a number (partial df/partial dt) at t = 0, the point of definition of the vector. So we imagine both the function f and the point t changing to yield different numbers (or different functions of t on different points t on the curve.
page 56: 'The mathematical model we shall use for space-time ie the collection of all events is a pair (M, g) where M is a connected four-dimensional Hausdorff Cinfinity manifold and g is a Lorentz metric (ie a metric of signature +2 on M.'
Two models (M, g) and (M', g') are equivalent if they are isometric . . . we are working in an equivalence class of models.
Transfinite numbers take some of the certainty out of analysis, since the alephs are influenced by structures beyond themselves, This sentence has meaning, for instance, because it represents a certain point in the space of all meanings.
We seek a fully circular definition of the world or a duality AA-1 - I
Back to gravitation: what is a null geodesic in the transfinite network? A transformation requiring minimal computation.
[page 278]
We proceed by finding correspondences between features of the model and features of reality, ie the model is a reference frame for reality [as any culture is]
Hawking and Ellis page :
60 postulate a) local causality
61 postulate b) local conservation of energy and momentum
77 postulate c) Einstein 's field equations.
page 78: gradient corresponds to the second derivative of the metric.
One explores the spacetime manifold by travelling along curves through it (like my life or the path of a spaceship, photon or planet. [does a 'free' particle have any choice in its path through Einstein space-time? No, it is a classical deterministic theory (with singularities though)]
graviton
photon
A photon is a flattened version of the structural differences of two energy states of an electron. If received by an electron with similar energy structure available to it, it will move the second 'up' as much as the first electron came 'down' (but Doppler/gravitational shifts can change affect this). They act as noise in the communication of 'peer' electrons. Some electrons in a continuum state can handle a range of photon energies.
page 117: 'Any spacetime metric can in a sense be regarded as satisfying Einstein's field equations * because having determined the LHS of * from the metric tensor of spacetime, one can define Tab as the RHS of *. The matter tensor so defined will in general have unreasonable physical properties: the solution will be reasonable only if the matter content is reasonable.'Why am I doing all this. To integrate myself into my Universe. To resolve the contradictions I feel when I get up in the morning and look at a beautiful landscape and see how it has been compromised by the work of myself and my forebears. To work
[page 279]
toward spiritual fulfillment with minimum detrimental input to the system of the world.
#Transfinite dynamics - we replace the infinite concept of continuity by a chain concept where discrete links nevertheless make a continuous entity because their ends are topologically bound, rather as the subroutines of a computation process merge into one another imperceptibly by the process of calling and execution which at the level of the processor is the same no matter what size the 'block' of software that the machine is entering. [recursive function theory]
We might attribute the relative stability of the world to the idea that something happens (eg a photon is emitted) only when a Turing machine halts, whereas we see that only ℵ0/ℵ1 of all possible Turing machines halt.
From a software point of view, the execution of a whole Turing machine is identical to the execution of one instruction in a Turing machine. Ie the dynamic process of computation is scale invariant.
All of the above to # is the result of lying an extra half hour in bed wondering if any progress at all is possible in this project. What I say to myself is that from a dynamic point of view, my tiny activities are the same as the works of Einstein, the Pope of the president of the US.
Further that the search for divinity is not so much a question of output (so valued in a productivist society) as simply of the pleasure of existence. God has no output. Everything for a god happens inside it. Here we see that that limit expressed by the fraction ℵ0/ℵ1, output over internal processing, which is very close to zero, in fact is zero in the system measured by ℵ0.
Friday 2 May 2003
Measurement: We preserve the past (and so are aware of it and can remember it) by taking a 'snapshot' so that a situation at time t0 is preserved in a system that does not change its state for t > t0 (having decided to model time by the convention t(past) < t(present) < t(future)). From Newton's first law, nothing changes without a force, we conclude that a static zone is a force free zone, or the forces are balanced in whatever way (algebra) necessary to arrive at no net force. So a written document may last for a long time, but eventually environmental forces eat it away. This is a statement of the fact that a lot of little changes add up to a big (and apparently continuous) change. At some point the papyrus becomes unreadable. But this point is also a function of the reading method as well as of the decay process, so that fancy techniques may extract information where a human reader can see nothing, eg UV.
There are two revolutions of the twentieth century centered around measurement. Einstein in the classical world of rods and clocks realized that the finite velocity of light changed the appearance of things moving relative to the observer. He characterized that change as the Lorentz transformation in a space with the metric eta. With the inverse of this transformation, one can transform the apparent changes away and see that the laws of physics are the same in every frame. it is natural to say that the 'real' physics is that observed by the local observer [and not by the distant bureaucracy!]
The apparent differentiation observed in other inertial frames is a result of the method of measurement, which depends on light signals to see the coordinates of certain points. So when observing an extended object at a distance, light may have to travel different distances to the observer from different points on the moving object. Different points on the object will then be seen as they were at
[page 281]
different times. Since the object is in relative motion, this means also that they are being seen at different places and hence the 'distortion' we observe. A similar argument holds for the clocks on a moving object, since at each successive observation it will be at a different place and so at a different distance and at a different delay. The net effect is to make moving clocks appear slower.
Quantum mechanics also revolves around new ideas of measurement. Here the governing principle is that there is a minimum quantum of action associated with changes in the state of motion of any system. A measurement is an action. Since the minimum action of measurement and the minimum action needed to change a system are the same, any measurement is capable of disturbing the system being measured., so it need not be in the same state after the measurement as it was before it. Just as a special theory of relativity can be built around measurement velocities ranging from 0 to c = 1, so a quantum mechanics can be built around a quantum of action ranging from h = 1 to ℵ0. In quantum mechanics we know, once we have made a measurement, what state the system is in, and we know it will stay there until it is involved in another interaction (measurement)
So in special relativity, the moving object appears to be in a different (always past) state than it really is. In a quantum system, each state is (for a certain time) in the measured state, but that measurement tells us the state that the system is now, but not what it was before the measurement. So the light behind the slit destroys the correlation of the electron coming through the two slits, so it no longer causes the interference pattern on the screen.
[page 282]
In physics we tend to separate measurement from all other activities, but in fact we could introduce an equivalence that measurement and action are the same thing and their effect is to establish some sort of correlation between two systems. In this way measurement is a communication. I ask a piece of wood how long it is and a measurement tells me. If I am in motion relative to the piece I am measuring, I must take the Lorentz transformation into account. This transformation is part of the measurement process. We can say generally (in line with the scientific method) that what we measure depends on the method of measurement. This holds as much for complex biochemical assays (like sequencing genes) as for simple physical measurements.
We try to get clues to a general model of communication in the world from the physical world. Classical physics works in terms of measurements of mass, length and time and relates all of these through the intuitive concept of force. In human terms, all measurements are made in terms of feeling, one feels a force and reacts to it. I feel hungry, soon I will eat. The correlation between feeling a force and acting is often quite weak. One can measure the strength of feelings by the rate of action they induce (rare, frequent, intense), just as one measures forces by the change of momentum they induce. One wishes to find a model which unites physical force and human force. This we hope is the formalism of formalism that we call the theories of communication and computation. We model the world as a network of computers, and apply this model at all levels. What we call a computer is a network of computers (gates) and each gate is a network of quantum events. Moving in the different direction, we ourselves and our communities are networks of networks of communication. This practice of extracting symmetries has a small world effect by making long distance connections between previously isolated communities of realities and researchers.
[page 283]
The rates of communication between different people in the world vary widely with time constants ranging from centuries to seconds. The fast communicators have a strong advantage over the slow ones and so are subject to an overwhelming tendency to exploit the slow, destroying their forests and taking their oil without compunction under the moral banner 'business is business'. How can we gainsay this. What is the transformation between systems (ie ways of life) that makes explicit the local standard of human existence? This is to be expressed in terms of communication theory, and is a constraint on communication protocols, eg whether one interacts with another with a gun or with full humanity.
We can think of the laws of physics as constraints on communication or communication protocols. [Since measurement is communication, we cannot expect to make a measurement that does not obey the rules of communication] Relativity and quantum mechanics deal with cases of maximum velocity and minimum action. How do we extend this physical law to slow processes with very complex action? The hierarchical network model links fast and simple to slow and complex.
The relationship between these two types of process can be seen in a spacetime. Time is as usual, but space is not measured with the usual Euclidian metric, but with a metric of complexity which relates (say) my complexity to the complexity of an atom. So we interpret the transfinite network as a spacetime in which special and general relativity happily fit. Then we model it as a space of action, thus obtaining a fit to quantum mechanics. This achieved, we then apply the theory to general evolution and see that complexity correlates with fitness and survival, arriving finally at the idea that the highest entropy society is the most durable. Finally, as a result of an action principle we want to elucidate the conditions for high entropy, ie the degree of order that maximizes entropy. [entropy has a cost, just as energy has a cost]
By studying the generally smooth large scale dynamics of the world, we can learn how to smooth out our own lives.
Hard v soft in social outlook may show in the relative levels of censorship of sensuality and violence. Both have their dangers, but where is the balance?
The chain notion of continuity is illustrated in manifolds by the overlap of coordinate neighbourhoods. In the overlap area the coordinates in one area are Cr functions of the coordinates of the other neighbourhood and vice versa. Tp be Cr a function must have the equivalent power xr, since dr xr/dxr = r!. For Cinfinity we need infinite polynomials. Hawking and Ellis page 12.
One step forward is to interpret one peer class of the TN as a manifold. The effect of the TN is to reveal the quantum spectrum behind each part of the manifold, ie what was once a line segment (represented by a metric) becomes a Cantor Universe in just the same way as 0 - 1 did under Cantor's analysis of points of accumulation (Jourdain in Cantor, pp
A transfinite network is a spacetime.
Once we have exploded a piece of natural line into a Cantor Universe inhabited by quantum theory, we have to see how operations like differentiation and integration in a manifold and elements of the manifold's organization like vectors, tensors and connections are realized in the transfinite domain which is connected to them and explains them.
Manifold: a set of points with local similarity to Rn - smooth addressing but integral dimensionality, so that each address is an ordered set of n numbers (usually considered to be reals), ie we number the dimensions with the natural a, but parametrize the dimensions with the reals. The use of the reals is justified because the theory of the metric involves Pythagoras
[page 285]
theorem and so square roots which may not be rational numbers. This makes mathematical sense but in real life we can only measure parameters to a certain accuracy, so that all measures of parameters may be adequately expressed with rational numbers which may be coded into naturals. So our manifold has a natural number of dimensions (ℵ0) each of which has a natural number as a parameter. This coordinate system can address ℵ1 points.
It can if we devise a suitable ordering system (similar to the way we write numbers as an ordered set of numerals) be used to name every point in the continuum, ℵ1. But this continuum, by its mapping to the manifold, must be considered a set of discrete points, It can be used to construct a set ℵ1 linearly independent axes, each with a parameter running from 0 to ℵ1, so that we can represent (encode) ℵ1ℵ1 = ℵ2 points.
Note that it is only in infinite (countable or greater) dimensional space that the transfinite transition takes place and this even if the parameter on each axis is only two (binary) valued, since
[page 286]
What is the meaning of this? It illustrates the power of exponentiation, but exponentiation is only possible with order (and also the invention of zero so that we can write 1[0]n to represent 10n.
Hawking and Ellis page 11: 'The only concepts defined by the manifold structure are those that are independent of the choice of coordinate system.
To set up a coordinate system for a set of n objects, we need a structure that can generate n addresses. These addresses may simply be aggregates of ones, in which case we need sum i = 1 to n objects , ie 1/2n(n+1) of them. On the other hand, we can take the dimensional approach where we have a dimensions and can address n objects with about n log n digits, ie to represent 4 things with matches, we need 1 + 2 + 3 + 4 matches, whereas we need only 4x2 = 8 binary digits. To represent 16 things with matches, we need 4 x 16 = 64 binary digits or 136 matches. This is the advantage of multi-dimensional addressing. What is the disadvantage? The overhead of keeping order. Do these optimize in 3 or 4 D?
The entropy of the points is the same as the entropy of the space they are in. Each dimension has a certain entropy which depends upon the range of its parameter H = log x, so that the total entropy of a space with finite parameters and n dimensions is nH. For infinite parametrizations, however, Cantor found that the number of points in nD space is the same as the number of points in a line, 1D space.
[page 287]
Transfinite arithmetic is strangely featureless, only transfinite exponentiation, logarithms and roots getting us anywhere. . . .
Maybe it is this very featurelessness that a) makes it 'soft' and capable of dealing with more humane matters and b) models the exponential structure of time, growth and decay.
. . .
The bases of a manifold [M] are different ways of mapping the points of M to ordered sets of reals.
The Cantor Universe gives a way to name every point in the transfinite manifold where we think of each set of a certain transfinite cardinal as a 'dimension' of the Cantor Universe, and the dimensions are all orthogonal in some way which we have to work out. Although general relativity only involved reals (?) perhaps we have to put complex numbers in somewhere so that the transfinite network can hold quantum mechanics.
[page 288]
We begin with the physical concept that the items we number with the natural numbers are quanta of action. In other words we recognize ℵ0 different quanta of action, each being a halting Turing machine. We seek support for this claim by the formal machinery of quantum mechanics and quantum information theory. So we turn to Nielsen and Chuang. Nielsen and Chuang.
We would like to see how these quanta of action behave in first an harmonic oscillator and then a transfinite oscillator. Let the initial singularity then be the superposition of all the energies of oscillation nh, n equivalent to ℵ0. This harmonic oscillator is now present at every point in the Universe due to the big bang creating a forest of different oscillators This forest is to be represented by the transfinite oscillator.
The initial singularity is a superposition of bosons with no spatial extent and so no special relativity only time. Now how do we get fermions, spacetime etc etc? Is there something transfinite about it?
Also since no spatial extent, no momentum or angular momentum, just energy which is its rate of action integrated across all frequencies.
Maybe we have got angular momentum in the initial singularity like an electron or a photon but no spatial extension yet., so no linear momentum. Think of it as a vast superposition of photons with nowhere to go. How do we make somewhere? How is the symmetry broken?
Whatever, we have three descriptions of the initial singularity, Cantor Universe, quantum and relativistic. At this point time has no direction, so we can see what is happening as a superposition of 'positive' and 'negative' t.
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Saturday 3 May 2003
We can say that the subsequent expansion of the Universe is due to a) 'Cantor force', b) 'time reversed black hole' or c) quantum uncertainty. To get somewhere we wish to exploit these three explanations to throw light on one another, subject to the conditions a) there is no 'outside' influence on the initial singularity b) energy is conserved and c) internal consistency is demanded in some way.
My intuitive feeling is that we should start with spin, ie there is a force inclining me to read Sin-Ito Tomonaga The Story of Spin again. Tomonaga So here goes. This force is the gradient of a field along the Tomonaga direction. I am also tending toward Feynman which is easier to understand. Feynman
The Cantor Universe is a 'recursive' reference frame, so we can start it wherever we like and it will work for the world if it too is a recursion. The 'size' of the starting point may be anything from the initial singularity to the origin of the earth, to any other point in our recursive Universe.
The first fit is at recursion = creativity.
How do 'spin' and recursion fit together? Time reversal reverses spin direction. A function takes a complex system and maps it into a number. A phone wire is [modelled by] a function mapping every phone to one [one phone to another?]. There are as many such functions as there are [pairs of] phones in the network.
It is a big puzzle and one cannot tell for sure whether one has got any of it out until one has got all of it out because the shape of the pieces depends on the shape of the whole. This is where reality comes in because we can see the shapes of the pieces, although how they are talking to one another is a much more difficult question. In a quantum harmonic oscillator made of photons and electrons, the bound states of the electrons serve as frequency converters, allowing (in the limit) any frequency to be converted
[page 290]
to the sum or difference of any others. This plus the frequency converting effects of general relativity give us [over time] smooth access to all frequencies, ie atoms have discrete energy levels which are slightly fuzzy in practice and give a certain line width, and gravitation [and relative motion] allows smooth conversion from one frequency (energy) to another.
In the initial singularity all energy is concentrated on a curve we may call time. We might describe this by a one dimensional general relativity whose stress energy tensor in one number h.ℵ0, the energy of the Universe, and whose spatial curvature is also ℵ0. What does this mean? Since our curve is not embedded (so far) in any higher manifold, and since we have not got a direction for time, we might think of the initial singularity as a small circle, radius 1/ℵ0 around which energy is running in both directions. From a formal point of view, we might say that any ordered set constructed in this system ie deconstructed by its inverse. To capture the notion of this inverse, we may find a place for complex numbers. We are thinking, however, that the quantum of action is already in place and that our little Universe is a superposition of frequencies and energies and intervals of time which cover all values from 0 to ℵ0. At present we explain this structure by saying what else could you expect in a Universe confined only by logic? Where did all these numbers come from? In processing terms, how do we implement Peano's axioms? The answer is in the bootstrap: the transfinite numbers are themselves a model for the natural numbers.
'cardinal of' is a function which counts the object in a set (using some sort of pattern recognition to decide what is an object, and to decide if this one has been counted before) and puts out a number, the cardinal number of the set (defined by the recognition principle). eg how many records with specification S in this data file?
Back to spin, whose strange properties entered our world through the Stern-Gerlach experiment. Michael Betz
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How does the world build itself? By doing everything which is not inconsistent. Boltzmann's law gives us a way to go from energy levels to probabilities. The derivation of this law depends upon ordering and permutation.
Agatha Christie The Hound of Death. The Call of Wings. manic-depressive? heaven=hell? ecstasy_hangover? Christie
Christie page 142: 'None of these explanations will do' . . . 'There are others, but they're not generally admitted.
Our heuristic principle: only the impossible is rules out, and impossibility is structure dependent, so we will claim as a consequence that from any infinite starting point (ie ℵ0 we can find a route (geodesic) to anywhere.
The only way to get off one's geodesic is to be pushed (as the earth pushes me) but I push back, putting the earth too off its geodesic. But our mutual centre of gravity follows a geodesic, unless there is a non-gravitational interaction with other bodies, ie radiation pressure from the sun etc.
An harmonic oscillator is built on a duality, which we represent by a pair of mutually inverse functions that map one state to another and back. The states are essentia, the energy esse.
Cantor's proof. It is impossible for a power set not to have a larger cardinal number than the original set. (so we would never expect to see a smaller power set in real life, but there is no reason in the proof for the power set to exist. Mathematics says if.
Cantor's proof plus the theory of probability (based on certain methods of counting) gives us a force to create.
Urge to smoke another joint, since cannabis seems to free my writing and what I write seems to me useful in the service of my overall desire. The only inhibition is the life shortening effect
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of smoking. Then I think to myself that the likes of Einstein and Freud were heavy smokers and died of it, but perhaps it helped them to make it over various lines so as a piece of altruism their smoking was reasonable as well as pleasurable. Every such rationalization depends upon a (dubious?) choice of reference frame. Will wait a while yet. Thought comes as fast as it can be written down, which is fast enough. In a full world, every action involves destruction.
If there were no other forces in the Universe we would not feel gravity because thee would be nothing to push us off our geodesics, and so we would always be in an inertial frame. Other forces enable the construction of extended objects in which tidal forces become apparent, as we would feel if we fell into a black hole.
Our first triad, action, time, energy makes no reference to space. Intuitively, however, spin does, both through our image of something moving at a certain radius and the angular momentum axial vector.
We may think of the initial singularity in two ways, either as a point isolated from the spacetime manifold as Hawking sees it or as the 'dust' from which the spacetime manifold is made by establishing relationships between the dust particles which both distinguish them from one another and give 4-space the structure suggested by Einstein's equation.
Our construction of the Universe may serve among other things as a pedagogical route to explaining the very complex systems (and the mathematical theories thereof) in which we now find ourselves. It would be sweet if we find that the Universe builds itself rather in the same way as mathematics builds itself, from primitive to complex notions. This building process is (creative) recursive function theory which we feel is closely allied to the slow development of our minds from infancy on. The structural growth of mind (like everything else) being realizations of the symmetry captured by the transfinite oscillator.
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Cantor discerned infinite structure in the natural line. We identify that line with the time line of the Universe and see the expanding Universe as an image of the structure of the lines revealed by Cantor.
Particles (and spin) arise from the constraints of special relativity (Weinberg). How? read read. Weinberg
. . .
The Cantor Universe is an attempt to build a reference frame for the natural line. The question is 'do we find the transfinite cardinals in the real world? According to physics, we can certainly find ℵ0 and ℵ1.
Christie page 156: 'It was his money that held him in bondage . . . . But was it? Was that really it? Was there a deeper and more pointed truth that he had not seen? Was it the money or was it his own love of the money? He was bound in fetters of his own making; not wealth itself but the love of wealth was the chain." High energy, low variety.
Comfort vs Call of the Wild.
General covariance - a rose by any other name would smell as sweet.
Initial singularity is a dust of Turing machines, ℵ0 halting, ℵ1 not halting. Gradually they become a network and all turn into oracle machines so that all ℵ1 halt and we have a spacetime manifold or network. [network can have addressing at all levels of smoothness].
The role of mathematics in physics is to provide paths between one set of measurements and another. In practice these measurements can be represented by the natural numbers. Given ℵ0 possible measurements, we have ℵ1 different mappings from [one] subset of the measurements to another.
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Fit 1: the big bang Universe.
As usual all this is very fleeting and I hurry so as to capture it. We are basically called on to do transformations in different transfinite domains. The business of science is to capture the world in writing. Writing is the definitive form of scientific archiving and communication and it is to the writing that we turn when we come into dispute. Despite this, the fact that we can speak much faster than we can write and publish means that only a tiny fraction of scientific communication appears in the literature. The literature is a distilled and abstract version of scientific discourse.
General covariance enters the picture when we try to write something down, but we can use the entropy of the system as an invariant as we translate from language to language (transform). So the Cantor Universe gives us domains of different entropy conservation, beginning with the natural numbers. The entropy of a message is invariant with respect to its representation in any language. So child raising is just as complex no matter what language parent and child speak.
To devise a universal translator (transformer) at constant entropy, we need to seek the maximum symmetry (minimum entropy) representation of the meaning in question in order to minimize the actual computations needed to perform the translation.
TRANSFORM @ CONSTANT ENTROPY = PERMUTATE
Getting a new lover while still married to the old can be an unstable situation, since it couples to the marriage, and is in danger of being disrupted, at some (unequal?) cost to the participants.
