Notes
[Notebook: DB 59 Draughts]
[Sunday 27 August 2006 - Saturday 2 September 2006]
Sunday 27 August 2006
Monday 28 August 2006
[page 134]
Weyl, Space, page 97: Riemann: 'The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metrical relations of space. In this question, which we may still regard as belonging to the doctrine of space, is found the application of the remark made above; that in a discrete manifold the principle or character of its metric relations is already given in the notion of
[page 135]
the manifold, whereas in continuous manifolds this ground has to be found elsewhere, ie has to come from outside. Either therefore, the reality which underlies space must form a discrete manifold, or we must seek the ground of its metric relations (measure conditions) outside it, in binding forces which act upon it'. Weyl eg Shannon's theorems.
Weyl page 98: '[Riemann] asserts . . . that space in itself is nothing more than a three dimensional manifold devoid of all forms; it acquires a definite form only thought the advent of material content filling it and determining its metric relations. There remains the problem of ascertaining the laws in accordance with which this is brought about.' eg the 'shape' of communication.
'We shall illustrate in greater detail the bold idea of Riemann concerning the metrical field produced by matter and we shall show that if his opinion is correct, any two potions of space which can be transformed into one anther by a continuous [logical] deformation must be recognized as being congruent in the sense that we have adopted, and that the same material content can fill one portion of space just as well as the other.
'Beginners are often confused by failing to notice that in the mathematical literature symbols are used throughout to designate functions whereas in physical literature (including the mathematical treatment of physics) they are used exclusively to denotes "magnitudes" (quantities)'
In physics, f(t) is an 'equation of motion' usually assumed to be continuous. We may imagine an f(t) which describes the discrete sequence of states of a Turing machine, beginning with the input and finishing with the output.
[page 136]
't' is the 'program counter'.
Tuesday 29 August 2006
Wednesday 30 August 2006
Top down, vs bottom up. Bottom up is linearize by differentiation and then integration. This has been an enormously fruitful strategy right up to the integral gymnastics invented by Feynman to give us the path integral method in quantum mechanics. Feynman The weak spot in all this machinery is the use of infintesimal quantities? Do they correspond to anything in nature? On the face of it know (sic). The Universe that we observe is the quantum Universe, which comprises a set of discrete events measured by Planck's quantum of action h bar which may be combined in an infinity of ways to give the Universe we experience. We model the behaviour of all these particles (which include ourselves and the planet) with quantum mechanics whose central mechanisms are differentiation and integration which involve the notions of continuity and infinitesimality,
Here we introduce a discrete mechanism to explain our experience of life. To arrive at this mechanism, we adopt new definitions of continuity and the process of 'passing to the limit'. Let us understand 'logical continuity' as the condition for valid inference according to the rules of predicate calculus. In place of a limit, we define a minimum event or process whose execution requires one quantum of action. We model different instances of such an event by Turing machines. Since the set of Turing machines is countable, we expect the set of
[page 137]
quantum events also to be countable. In quantum mechanical terms we predict that the set of eigenvectors of the Universe is countable.
This we take to be locally true, where by local we mean a point corresponding to a Turing machine with certain software that transforms a set of inputs into a set of outputs. A Turing machine is a function (like sin x ) which produces the number sin x every time we feed it an x. Sin x can deal with real and complex numbers, but does not have any way (as it stands) of dealing with vectors and more complex structures without itself being embedded in a more complex structure like vector space.
To become common currency an ideas must be represented in a functional (working) way.
Invariant theory
We begin connecting the symmetric network to the observed world by considering the special and general theories of relativity. They identify certain features of the Universe that remain the same no matter how much everything else changes. In this way Einstein furthers Parmenides program which was to explain everything in terms of an eternal, perfect, totally invariant entity. God so understood serves as a point at infinity that give meaning to other points.
Einstein derived the group of transformations of
[page 138]
special relativity (the Lorentz group) from one principal idea: every freely moving observer see the same Universe, including the same velocity of light. This is a local theory. In a Universe where continuity is defined by inference, seeing means receiving information from one's environment. So, moving from particle physics to molecular biology, neurophysiology and so on, we can trace the logical journey from photons leaving certain atoms to the proposition (spoken or not) 'I see a table'.
The Lorentz group results from the delays in the transmission of information from one inertial frame to another in relative motion. The result is the well known effects of time dilation, contraction, mass increase and so on induced by the interplay of relative velocity and the finite velocity of information. When we observe that the lifetime of a fast moving cosmic ray particle is longer than the lifetime of the same particle at rest in the laboratory, we can use a Lorentz transformation to calculate the velocity of the cosmic ray. An observer in the rest frame of the particle would see its rest lifetime. Nevertheless the photon, travelling at c, has no rest frame. In all frames its observed velocity is c.
The Lorentz group is a continuous group in classical space, and we can study it by the time honoured method of of integrating a series of infinitesimal transformations, each corresponding to an infinitesimal change in the relative velocity of the inertial systems under consideration.
The velocity c is the maximum rate of transmission of
[page 139]
information in the Universe and so has cosmic significance. But every communication process has a characteristic velocity which is a local parameter. So we can construct a set of light cones and transformations for information travelling by sailing ship as we do for information traveling by photon. This circumstance gives causal structure to shipborne trade in the same was as c gives causal structure to the Universe.
To explore the application of special relativity to the symmetric network, we begin with an informal discussion in terms of human communication networks, which are familiar to us all.
Since we conceive of the Universe as a network of networks, each of us may be considered a network whose input and output is our interactions with other in our peer group. All these interactions go down through the layers of the individual to the physical layer of one individual whence they are transmitted through the physical layer of particles to the the individual whose system transforms them [back] to humanly meaningful information.
Collectively we call the representations of permutations in the symmetrical network 'transfinite symbols'. These symbols vary in cardinal number, but we assume that there exists a transfinite symbol complex enough to be placed in one-one correspondence with all the processes in myself, including communication with other people corresponding to other transfinite symbols.
Now to communicate is to correlate, but how do we reconcile correlation and no cloning? Here spacetime enters by making
[page 140]
it possible to have identical particles (like electrons) that are nevertheless differentiated by different spacetime 'addresses'.
We explain the delay in communication by saying that all communication requires encoding and decoding which requires logical processing by Turing machines which takes time. Higher energy = more actions per second = faster processing = less time for a given task.
The harmonic paradigm integrates all its processes down to a countable magnitude which a physicist can observe.
HARMONIC PARADIGM - RECURSIVE FUNCTION THEORY.
The symmetric paradigm gives us a new foundation for counting and integration which imitates the harmonic paradigm in the limit of large numbers. The harmonic paradigm itself is the large number approximation of the symmetric network. It calculates the traffic in the symmetric network without telling us the meaning of the traffic, rather as a complete dissection of global cashflows tells us the traffic in value, but not what people are actually buying and selling.
Lorentz transformation gives us a local view of spacetime based on the invariance of local experience. The foundation of general relativity, general covariance., shows us how to synthesize the local values of every particle in the Universe into a global view of the whole network (at any peer level), including that characterized by c, maximum velocity, minimum complexity, constant action (?)
[page 141]
Can we say that instantaneous action at a distance contradicts the notion of distance and would reduce the Universe to a system of complexity zero, equivalent to Parmenides model of being?
Human communication: we assume that each human personality is represented by a transfinite symbol. The 'relative motion' of these symbols is encoded in their differences. In other words, we interpret transfinite symbols dynamically as motions that is networks of Turing machines. The relevant Lorentz transformation transforms your personality into mine so that I can sand in your shoes and so see our common humanity.
To go further in this direction, we introduce a permutation interpretation of general covariance, starting from Riemann's geometry,
Since my views are heretical in the traditional theology that dominates society and academia, I am theologically homeless, so knocking on the door of the scientific community looking for a living. But a bit too wild and vague for that so stick to the entrepreneurial mode : technology leads science by visions of practical payoff.
In quantum mechanics all physical measurements amount to counts. We bin counts in different ways and seek to find and explain relationships between the resulting numbers. The computer network connected to the Large Hadronc Collider and the neural network attached to my eyes transforms raw counts at different transducers into something meaningful to a user. I see a Higgs boson; I see a table.
Thursday 31 August 2006
The metric properties of spacetime depend upon its momentum content Tij. By momentum content we mean the spacetime detail (entropy) of its structure (processing rate). This is the key. Expand on this to see the whole arch.
4-momentum = 'processing density' (Paris university c13)
Processing density attracts processing density ('brain drain')
The judgments 'good; and 'bad' are local relative to the local frame of reference (ego). So I say good for me, bad for me. Then invoking the symmetry of humanity, try to extrapolate this to a world which is good for all the egos it contains.
The sum of all human flights of spiritual fantasy is strictly greater than the spiritual fantasies located inside a given ego (persona). Various people and institutions try to impose artificial symmetries on human noetic space, various forms of mind control which are best implemented by upbringing and education. These constraints are imposed from without. By applying the ideas of general covariance, we can draw the line between 'forced' and 'free' education, and so implement the free (generally covariant) algorithm of life.
Conservation of energy and momentum are not automatic, but the orthogonality of elements of the Universe is maintained by feedback (error
[page 143]
correcting) mechanism. Feedback negative e-x, positive e+x, x is a real. Or oscillation, x is a complex number.
Symmetry: only be different for a reason, ie some payoff. Breaking a symmetry carries a cost which we model as special processing. So vegetarian meals on aircraft. Rich and poor is an unnecessary (costly) break in symmetry.
But what if the potential is so shaped that breaking the symmetry releases energy as it increases entropy? The potential shapes things to a certain extent (eg gravitation) but a lot of possibilities remain all taking gravity into account in an effective manner, including various forms of structural strength, ie meeting potential with potential. So electromagnetism maintains a certain level of structure n the face of gravitational potential, and stronger forces appear with increasing momentum density, till finally gravitation triumphs as we are left with a black hole, a structureless structure.
Imagine q, imagine 2, imagine 3. . . .
Now imagine the transition from 1 to 2, imagine the transition from 2 to 3 . . . Now imagine the set of these transitions as a lot of units which can be added up and we have a cardinal measure of the size of the set of natural numbers, or any other set of numbers.
The harmonic paradigm is an implementation of the law of large numbers. By taking inverses of everything, epsilon-delta arguments can be written as 1/epsilon, 1/delta arguments. As the size of the increment goes down, to the number of increments goes up, but they all remain units, cardinal numbers that can be
[page 144]
added and normalized. When we say integral f dx the infinitesimal dx neutralizes the increasing number of 'samples' of f that we are adding together.
Now the Universe is not all large numbers. Locally it is [quite small].
Point in spacetime ('event') only has meaning insofar as it communicates with (operates on) another event. Cosmic process is a 'flow' of events, local past influencing local future through local present.
Friday 1 September 2006
Saturday 2 September 2006
Time to get some numerical values into all this.
Why quadratic forms *inner products etc)? Because communication between permutations is bidirectional?
Numbers measured by physics are cardinals, but we can extract ordinal numbers by measuring cardinals at different pints in spacetime by different methods, ie using different ordered frames of reference. [probability theory gives us a fundamental cardinal frame of reference Kolmogorov]
Distance between permutations = number of substitutions required to make them identical, ie Hamming distance. Minimum 0, maximum ℵ0, at the countable level.
How do we account for null geodesic? or, more particularly, -1, 1, 1, 1 diagonal of eta(ij)? 'signature of metric'. In spacetime distance is 0 for massless particle traveling at c. What
[page 145]
does this mean in terms of sailing ship communication between England and Australia? d1 a posts a letter; dn b gets it and replies; d2n a receives reply.
Combination = permutation in which some of the symbols are indistinguishable in themselves (identical) from the pint of view of the combiner. Nevertheless distinguished at least by spacetime coordinates. [4-D = minimum complexity for effective communication?]
The size of the symmetric Universe is such that for every x it contains not-x. This holds particularly for the initial singularity (is ), which has no meaning until it has grown to is + not-is. [since by definition even the operation not cannot take us outside the Universe, the only way to go is 'up' or 'forward in time' to more complex structures within which not can operate.
Maximum distance is ℵ0 at the countable level, but there is no upper limit at higher levels ℵ1, ℵ2 . . . Nevertheless the symmetric structure gives us wormholes, ie knowledge or symmetry. eg consider the permutation
1: abc, 2: acb, 3: bac, 4: bca, 5: cab, 6: cba considered as a single symbol whose elements are the permutations of three elements a, b, c. Now we permute the basis, a, b, c so that a --> b, b --> c, c --> a.
Then we have 1 -->4, 2 --> 3, 3 --> 6, 4 --> 5, 5 --> 1. 6 --> 2, a permutation 'induced' by a permutation at peer level -1, pl -1.
Now we can make a distance matrix
[page 146]
|
|
abc |
acb |
bac |
bca |
cab |
cba |
|
abc |
0 |
2 |
2 |
3 |
3 |
2 |
|
acb |
2 |
0 |
3 |
2 |
2 |
3 |
|
bac |
2 |
3 |
0 |
2 |
2 |
3 |
|
bca |
3 |
2 |
2 |
0 |
3 |
2 |
|
cab |
3 |
2 |
2 |
3 |
0 |
2 |
|
cba |
2 |
3 |
3 |
2 |
2 |
0 |
This little world has no distance = 1, only 0, 2, 3.
We may consider the distances as times, measured by the number of swap operations the Turing machine has to do.
'Induced permutations' are invisible to the peer level than induces them. So If I steal somebody's book, ie permute the book from 'not mine' to 'mine' I cannot say what the overall effect on the previous owner will be. This gives also a glimpse of covariance.
'The world of logic'
The basic parameter in physics seems to be proper time, ie the local count of clock ticks (processing operations) between events.
Heisenberg: page 2: 'A much more radical departure from the classical conception of the world was brought about by the general theory of relativity, in which only the concept of coincidence in spacetime was accepted uncritically.' Heisenberg
All communication occurs by coincidence, ie by points on
[page 147]
the 'velocity cone' (= light cone when v = c ). So when I send you a letter, as far as the letter is concerned, the point at which I write it and the point at which you read it are coincident. Like a photon, a letter has no 'rest frame', its motion cannot be transformed away and so must be some sort of 'universal constant'.
The graphic user interface of the Universe is pixellated, but they are 'tensor pixels' whose size and frequency vary with local 'density' of 4-momentum = 1 + 3 momentum.
The physics of any particle is relative: not relative to any coordinates, but relative to its environment. [relative to any observer] Each particle observes its environment in a time division multiplexed way and synthesizes this picture into an abstract foundation (representation) of strategies for survival.
Recursive function theory: Turing machine = {Turing machine}.
