Notes
[Notebook: DB 55: TURKEY]
[page 253]
[Sunday 20 April 2003 - Saturday 26 April 2003]
Sunday 20 April 2003
Easter Sunday: an ancient occasion of joy somewhat corrupted by the bizarre tale which is used in my heritage for its context. This is a heritage driven interpretation of spring!
Heritage: this is the general term for everything one acquires from the environment into which one is born.
A tensor represents the process by which a complex structure (also represented by a tensor or a vector) is reduced to a scalar for communication purposes. The value of the scalar constrains the reconstruction of the complex structure in the receiver of the message. If enough scalars are transmitted in order (time ordered vector) the output of the channel can be fully constrained by the recipient. So tensor = coding. Because this communication process is reversible, we know
[page 254]
that the information in the structure and the information in the message have the same mass.
The traditional mathematics of physics is worked out at the interface between the natural and the real numbers. Details of the transitions from N to R will be found in any text on analysis. Here we want to suggest that while that space gives us a firm grasp of the relationship between discrete and continuous features of our world, we need to broaden the mathematical stage to capture the full complexity of ourselves and our environment, here the transfinite numbers which apply the countable/uncountable transition recursively to generate an expanding and evolving Universe. Transfinite network. Hille
In the network context we are able to discuss the question what does a message mean? What does a particle mean? Here we must make the N - R transition again, because the meaning of one bit of a particular message can only be determined in the context of the whole network. Here the transfinite approach helps because our model has a symbol for every conceivable bit of information in the Universe. Having identified every point in the model with a point in the world, we can then start looking for the symmetries of the model in the symmetries of the world and vice versa.
Tests:
special relativity
general relativity
quantum mechanics
quantum field theory
evolution
biology
spirituality
All communication is relative. We test by fitting to existing models.
Remember that Turing machines can be made with any different mathematical infinities, from 2 to ℵ0. The formula
2ℵn = ℵnℵn = ℵ(n+1)
tells us that within a cardinal number, actual size does not matter. The transfinite numbers are inherently digital.
The 'master equation' of transfinite arithmetic
(1 + ε) ℵn = ℵn, ε > 0 ?
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what is the lower bound on epsilon? 1/ℵn, 1/ℵn+1
(1 + 1/n)n = e.
There is a feeling of certainty about all this, but no proof. In a way, the uncertainty principle is built in, but we know that god, while subtle, is not malicious, and that we can know every detail of our existence, although the sheer number of details prevents us from knowing them all. The only memory big enough to hold the whole Universe is the whole Universe (as god's intellect is identical with god in the traditional theory, yet distinct as the world of god). The thing is a bit like a hologram, however, every point of which contains a low resolution version of the whole.
Do the large numbers involved in the transfinite network mean that all statistical quantities adhere perfectly to their 'classical limits' due to the law of large numbers?
The law of large numbers holds, but it is the fluctuations that are of interest because they are in effect fluctuations in counting conventions and so can cause themselves to have a greater probability of existing (by positive feedback) or a lower probability (by negative feedback).
Decomposing a permutation by power sets gives us an enormous new breadth of structure which we can now exploit with some sort of evolutionary algorithm that can lead us to the present world. In my mind I have forty years left to figure it out. As if!
Monday 21 April 2003
Probability and cycles. The probability of a random event becomes much greater if that event is part of a cycle, ie can draw energy to itself by bonding. So gravitation has two aspects - cosmic
[page 256]
expansion peppered with local contractions. [open and closed sets]
We assume ℵ0 halting Turing machines and see if we can map them onto the number of different operators present in the Hilbert space of elementary quantum mechanics.
MTW Misner Thorne and Wheeler.
MTW page 48: 'The idea that every physical quantity must be describable by a geometric object and that the laws of physics must all be expressible as geometric relationships between objects had its intellectual beginnings in the Erlanger program of Felix Klein (1872), came closer to physics in Einstein's principle of 'general covariance' . . . and today pervades relativity theory, both special and general.
Geometric object = ordered set.
Network is a geometric object. Fundamental network operation is sending a message from A to B, which has the effect of correlating A and B.
Point of view (from the point of view of the receiver of a message) is the algorithm (Turing machine) that one uses to decode the incoming message, ie find the MEANING in it. [for Turing machine we may read 'network of Turing machines']
differential = rate of data flow = rate of entropy flow
integral = total transfer of data
.page 131: At every event in spacetime there exists a stress energy tensor. It is a machine that contains a knowledge (representation) of the energy density, momentum density and stress as measured by any and all observers of that event.
A point is a Turing machine.
WORLD = river of 4 momentum running along Cantor Line.
[page 257]
MEANING: something one can act on. So I understand the meaning of mathematics when I can do calculations, ie 'see' how it works = see the essence ie be able to manipulate the object whose meaning is understood.
Peano Axioms yield the same cardinal number of natural numbers as there are Turing machines [?]
Theorem: The cardinal of the set of distinct Turing machines is
ℵ1
Theorem: The cardinal of the set of halting Turing machines is ℵ0
Execution of a halting Turing machine is an event.
Now apply probability theory.
If our mind was simple enough to understand, would we be too simple to understand it? Yes? No?
MTW page 139" Ideal gas has no off diagonal (stress) terms in is stress-energy tensor.
Although we have ℵ0 Turing machines, there is a distribution in their actual halting times, which we shall call the 'length' of the Turing machine. The number of machines available at each length is limited, so we have the origin of time structure in the lengths of Turing machines.
Now recursive function theory says that we can make big Turing machines out of little ones by a 'calling' and 'returning' process. Each subroutine is a process. So we may imagine an electron as an event, a subroutine that some in handy for making atoms, electron beams and so on.
For evolution we need randomization, hereditability and selection. Random exists because there can exist multiplicity and equality. Equal = not ordered. The cardinal symmetry.
CARDINAL SYMMETRY broken by CANTOR SYMMETRY.
[page 258]
Heritability depends on existing order which can be coded in an abstract form, eg DNA is an abstract version of a living organism.
Approximate models: 'assume a spherical chicken . . . '
The world line of the initial singularity is a scalar time. Consider this to be the real line and then apply Cantor's theory to it.
Tuesday 22 April 2003
The real learning is how to correct error condition, ie to get the system going again by reconstructing broken communication channels (eg the cam in the distributor is no longer opening the points sufficiently).
The communication of information always involves the communication [transportation] of matter (energy). So a good foundation for linking physics and information theory is the relationship between energy and entropy.
How do we say Einstein's equation in the language of the transfinite network? Here we say energy is modifying the structure of space and the structure of space is modifying the motion of the energy. We model this in the physical world by programming a Turing machine to carry out the necessary computations at each point in spacetime, in other words to implement the relevant tensors. These tensors are an abstract representation which have meaning in the environment of physics. How are they implemented in reality?
We postulate an evolutionary process within the context of the network. Why does the network assume a structure which appears to us who inhabit it as 4D spacetime? We want a) to explain what dimensionality means to a network and b) find a variational principle which is stationary at the current spacetime structure. This principle is a reflection of the stability of 4 -space, or should we say its fitness.
[page 259]
We begin with the initial singularity which is considered to be a point moving along a line parametrized by action. The only action is to perform some cyclic event with a complexity of 0. We have zero entropy from a spatial point of view. This motion, to have meaning, must be complex (ie relative), so we represent it by the function φ(t) = eit. We want this thing to replicate.
This can happen by the development of new phase relationships, so that we now have two streams of action superposed that on the 'collapse of the wavefunction' can give us two spatially separated (ie simultaneous but distinct) events. These two events represent two independent channels of communication (ie two degrees of freedom, space and time). Action continues through each separate channel, and different messages may be sent over each. The channels may nevertheless communicate with one another (rotation of axes) while maintaining their independence.
Think of this as a superposition of different modes of the universal wave function, giving different outcomes at different measurements.
Measurement could not begin until there was a duality, measurer and measured. Since we see each event (each tick of the aleph) as a measurement, we need some duality (before and after) to grasp it (ie make it communicable).
So communication correlates our action and space channels.
Given this, what are the costs and advantages of having more than one dimension, ie more than one machine operating at a given spacetime point? Parallel
[page 260]
processing enhances speed, and speed is of the essence for survival. To say that 4D space has a selective advantage is to say that it gives the best balance of processing speed against cost of maintaining independent channels, so this is how the momentum of the Universe flows, an echo of the trinity.
SEPARATE CHANNELS = SEPARATE MEMORY/PROCESSOR
Different bits of memory may share the same process, but different processes cannot share the same bits of memory without becoming entangled. To communicate, however, is to share memory. I put a symbol in your memory, you put symbols in my memory. We may see 3D space as the memory of the world, a place where things go slower than the processor. So we can conceive of the unity of the cosmos as one processor working with many bits of memory, and an event at a certain spacetime point is just the processor accessing and changing that point in memory. The flow of action in the Universe is constant, but where it happens is variable.
How do we understand Lorentz transformations, which change processing rate and memory so that dt2 - dx2 - dy2 - dz2 = ds2 = constant?
Why does this mixing obey Pythagoras theorem?
Why is God 4D and spatially extended?
Newton's first law: we create memory by not processing. Thinks stay the same until changed. So one processor working in a big space changing only one thing at a time leaves a lot of things unchanged. Given a processor running at ℵ0 in a space of ℵ1 points, each point gets changed at the rate of ℵ0/ℵ1, near enough to zero.
Logical confinement suggests that a cosmic Turing machine is a superposition of all possible Turing machines, each measurable event arising from a connection between two
[page 262]
of them. Why? Because thee is no reason to favour one over the other. All elements of this superposition, however, share the basic features of the Turing machine, however, which is that it is device defined by four parameters, ie 4 variables:
1. symbol read
2. internal state
3. symbol written
4. direction of motion
Making an animation - first the sound (effectively just one communication channel, working through time), then expanding into a spatial structure which changes through time in a manner keyed to the script, so that the abstract temporal representation is made concrete as a 3D thing, ie 1D sound and 2D coloured pictures. In the mind of the watcher, this is expanded in turn to an emotion laden interaction with the adventures of the characters.
Inversely, the emotional experiences of the writer are encoded in the written script.
One feels that all this is known. but putting it into words is difficult and after a while the energy for doing this seems to fade, and a rest is needed before the next session. This seems to be a general characteristic of work, (both reading and writing) that it exhausts the system a requires a period of recovery (sleep). Perhaps this is because we are inclined to work harder than our continuous rating, so that intermittent operation is required in order not to overload the system and cause too much 'temperature rise'. We tend to reserve high esteem for systems which operate in a continuous (failure resistant) manner, although such machines may not make as much progress as 'fluctuating' ones which make greater than normal effort at critical moments, ie when it seems possible to 'score' within the maximum effort available.
Nothing can 'stop' in 4-space, the river of 4-momentum continues forever . . .
MTW page 218: spacetime curvature = relative acceleration of test particles. Misner, Thorne and Wheeler, op. cit.
Acceleration = change of velocity = change of energy (both relative to something) so my car accelerates and gains more energy relative to the road network rest frame. So the relationship between the car and the road exhibits 'curvature'.
Since everything is relative, we can choose our 'fiducial' and 'test' situations in an infinity of ways. What we seek is a point of view that makes everything look simple (ie easy to understand and manipulate). In cosmology, this point of view is the inertial (ie 'acceleration free') frame.
MTW page 220: 'Spacetime curvature causes not only geodesic deviation, but also a route dependence on parallel transport (parallel transport around a closed curve changes a vector or tensor . . . ); it causes covariant derivatives to fail to commute . . . ; and it prevents the existence of a global Lorentz coordinate system.
page 221: Einstein: 'space was still [for physicists] a rigid homogeneous something, susceptible to no change of conditions. Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to to new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible.'
[page 263]
Each Lorentz frame is a distinct symbol in the writing of the Universe and communication between them transforms one into another by the acceleration described in the Riemann tensor.
The dynamic origin of Einstein's general relativity is the totality of all 3-geometries with positive definite Riemannian metric.
'Event' is identifiable, ie unique name or address.
MTW page 226: 'Vector' (ordered set) is local and must be attached to an event ie the address of the event is part of the specification of the vector. The same for 1-forms and tensors.
Because everything is relative, we can choose and construct coordinate systems which we use as a 'known' to describe the unknown. The transfinite network is a 'known' onto which we hang everything else. A good reference frame makes everything look as simple as possible, eg rotating axes to get rid of cross terms in the equation etc.
Round and round and round we go, so life is an almost periodic function but every revolution is different because it occurs at a different time.
The initial singularity is present at every point as a plane wave (local) of 4-momentum.
differential topology --> affine geometry --? Riemann geometry.
An important notion in the transfinite network is that any physical event may have an infinity of meanings, its actual meanings being determined by its positions in all the subsystems of the world that condition it.
MTW page 247: No local measurement that is insensitive to gravitational tidal forces can detect any difference whatever between flat and curved spacetime.
Wednesday 23 April 2003
Thursday 24 April 2003
[page 264]
Friday 25 April 2003
The strength of the network model is our intuitive understanding of it via gossip and so on. The structure of the network tends to spread information in different spatial patterns, depending on continuity. Baryons on the whole only talk to their neighbours, but [electrically] charged particles communicate via photons over vast distances creating the 'small world' structure. Watts and Strogatz.
Disasters are guaranteed in non-adaptive normative systems, because the gap between system and reality grows exponentially with the time between looks at the environment.
There is a chain of adaptation running in two directions, from the requirements of the parts constraining the behaviour of the whole and the requirements of the whole constraining the behaviour of the parts. So it is with people and societies. The peace theorem holds that there is a large region where both part and whole can have everything they want, leading to a peaceful situation. It may be that such a situation can only evolve in a static environment, given a long time, whereas in a rapidly changing environment the perfection of adaptation is reduced by the need for speed.
Networks are described statistically. When meaning is optimally encoded the meaningful symbols (semaphores) are equally probable, even if the input to a channel has a wide range of possibilities [E-theorem again]
Ambition and disappointment. The more impossible the ambition, the higher the possibility of disappointment but the greater the payoff for success.
Is it the exponential structure of probabilities (exemplified by the transfinite network) which makes control [knowledge and action] possible?
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The Cantor Universe is the space of axiomatic probability theory - Kolmogorov. Kolmogorov.
Quantum mechanics enables us to express two aspects of an event a) its structure (eigenfunction) and b) its probability (eigenvalue)
Saturday 26 April 2003
Pais 179: Einstein: 'I realized that all natural phenomena could be discussed in terms of special relativity except for the law of gravitation. Pais.
Special relativity = communication theory
General relativity: communication theory plus transformations between relatively accelerated frames, ie a particle evolves as it moves through curved space.
General relativity tells us how energy (= processing, communication) transforms space, and how space (essence, structure) moves (accelerates) energy. By seeing energy as a symmetry of communication, we expand the concepts of general relativity to apply to all process (represented by the term 'generalized geodesic') for the course of any entity through life (eg me) changing and being changed by my environment. So the Lagrangian from which we get General relativity represents a path through life which we can extremalise through variation. So we might see general relativity as the interaction of the sets ℵ0 (observables) and ℵ1 (first order spirit?)
Pais page 180: Equivalence principle: 'we assume the complete physical equivalence of the gravitational field and the corresponding acceleration of the reference frame.'
[page 266]
Flow of 4-momentum is a flow of spacetime detail [information, entropy]. Information is symmetrical with respect to representation, serial, parallel or mixed.
Bending of light = modification of text ie modification of representation.
Is information conserved on the way into a black hole? How does the entropy of a black hole compare to its mass? Ask Hawking.
We must think of everything in terms of measurements, ie relationships as in trinity and Einstein. Gravitation transforms all relationships indifferently, and can be seen perhaps as the structure enforced by the closure of a network so that there are no isolated nodes. Every node (including 'dark' ones) are alight to gravitation.
Pais page 195: There is no more absolute acceleration than there is absolute velocity.
Special relativity depends exactly in the delay in communication between spatial points. This delay is a metric . . . which, given certain assumptions, gives us a general description of space which is transparent to the actual content or method of communication, only that there is communication and delay. This is true of my life. It takes time for me to move from my point of creation to my point of annihilation.
First the equivalence principle, and then the construction of a theory of gravitation from which we can derive the equivalence principle.
Complexity, structure and transformation. 'Of course, one cannot replace an arbitrary gravitational field by a state of motion without gravitational field, as little as one can transform to rest by means of a relativity transformation all points of an arbitrary moving medium.
But by producing a function which describes an arbitrary moving medium using a fixed set of axes one can use the inverse of this function to neutralize the movement so described.
[page 267]
CAUSALITY <--> COMMUNICATION
Pais page 196: 'Einstein concluded his comments on the equivalence principles by stressing again the great heuristic significance of the assumption that it is true for all physical phenomena rather than for point mechanics only.
page 198: What and how can we measure?
Gravitational fields change the frequency but not the cardinal number of wave crests (or any other [observable] symbols) transmitted - the invariance of entropy/information.
page 201: [the laws of Euclidian geometry] most probably do not hold in a uniformly rotating system in which, because of Lorentz contraction the ratio of the circumference to the diameter of a circle should be different from pi if we apply our definition of lengths.
The place to work out the theory is in a 'transfinite manifold'
page 203: 'The equations of analytical mechanics has a significance which far exceeds that of Newtonian mechanics.
. . .
page 205: 'This is the dawn of the correct formulation of equivalence as a principle that holds only locally.'
How can I tell if I am moving myself, or are being led along by an idea? Locally impossible, but by measuring my (tidal) motion relative to my neighbours I can gain some objectivity. Our mental space may be able to be placed in correspondence with spacetime.
Tidal acceleration comes from talking to one another, so correlating and coming closer.
page 210: August 10-16 - Riemannian geometry needed.
[page 218]
10 gravitational potentials needed.
Pais page 211: Einstein: 'To throw out geometry and keep [physical] laws is equivalent to describing thoughts without words. What must we search for at this point? This problem remained insoluble to me until 1912 when I suddenly realized that Gauss's theory of surfaces holds the key for unlocking this mystery. . . . I realized that the foundations of geometry have physical significance. . . . So I asked my friend whether my problem could be solved by Riemann's theory.'
page 212 Einstein looking for the most general transformations that leave
ds2 = gmn dxmdxn
invariant.
This equation represents the immediate invariant quality of 1 - 1 communication = measurement.
page 213: 'gravitational field equations have to be non-linear since the gravitational field necessarily acts as its own source.
'Because of the Lorentz contraction in a reference frame that rotates relative to an inertial frame, the laws that govern rigid bodies do not correspond to the rules of Euclidian geometry. Thus Euclidian geometry must be abandoned if non-inertial frames are to be admitted on an equal footing.' Einstein 1921.
. . .
Physical laws require language = symbols (letters, numbers etc)
page 216: 'I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now. Compared with this problem, the original relativity is child's play.
4-momentum = bandwidth, a relativistic invariant.
[page 269]
Pais page 219: measurement: Einstein 'The gravitational field influences the measuring bodies . . . in a definite way.'
page 252: 'One may choose a coordinate system at one's convenience simply because coordinate systems have no objective meaning.'
Same for languages? Coordinate systems nevertheless have a meaning for the person who uses them.
page 273: 'the idea of general relativity is a purely formal point of view and not a definite hypothesis about nature.'
Let us get to know 'general covariance' in the transfinite network.
