Notes
[Sunday 2 August 2009 - Saturday 8 August 2009]
[Notebook: DB 67: jciii]
Sunday 2 August 2009
[page 77]
Monday 3 August 2009
A problem: if unitary evolution is continuous and meaningless why is it quantized? Answer: we do not observe unitary evolution (except through gravitation) and so we can say nothing about it. All we are saying is that all communication is quantized ('packetized') and we only observe that in spacetime and it is not necessarily reversible. The essence of lossless coding is that it is reversible, even if the computation process itself which encodes and decodes the message is not reversible.
How is memory generated? The unitary evolution of an isolated quantum system is controlled by the Schrödinger cequation which is a fixed point in the dynamics of the system, mapping itself onto itself. Schrödinger equation - Wikipedia Where is the Schrödinger equation recorded? How is it remembered?
From the point of view of algorithmic information theory the total information content of an isolated quantum system is represented by the Schrödinger equation which may be written in a few characters, i d psi / dt = E psi = H psi which holds for all values of H and
[page 78]
all dimensionalities of the vector psi. The differential equation is invariant with respect to complexity and energy. Completely undefined energy corresponds to a sharply defined moment in time.
Apply the information / entropy idea to the differential equation. Each new dimension adds a new symbol to the vector / word.
Cantor's theorem works insofar as the symbols in the proof are clearly defined. Insofar as they are vague, so the proof is vague.
PURE FORMALISM means PERFECT RESOLUTION, ie the probability of confusion is taken to be zero, ie the vectors are orthogonal. Each dimension represents the range of values of a given symbol, eg x the parameter which measures the distances on the x axis. Each discrete distance corresponds to some physical symbol, like an electron in the state psi, or should we say the state psi containing an electron (excited by an electron). Fermions have this property of orthogonality.
If this idea is true, perhaps we can associate each fundamental particle as a snippet of code which may be used with other physically embodied snippets of code to write a more complex system like an atom which may be seen as a fixed positive charge surrounded by an equal and opposite cloud of negative charge which can be localized on electrons.
As Thomas noted, our guiding principle in our study of God is to elucidate what God is not vs what God is, an intractable
[page 79] problem as we have learnt from the work of Cantor, Gödel, Turing and Shannon. Thomas. . .
Gravitation is the potential that makes it economical to shorten corridors of heavy traffic by moving heavy traffic sources (like black holes) closer to one another, ie MASSES ATTRACT.
How does the network model map to ordinary physics? The exchange of a minimal message is a quantum of action. The rate of communication on a channel is measured by its energy E = hf, where f is the time rate of exchange of messages. Energy = heavy traffic attracts energy (heavy traffic) tending to shorten the route between them. This only makes sense if long routes are more expensive than short routes. In telephony, this is becoming less and less true as the cost of the nodes grows relative to the cost of the links, wires, fibres and radio. How does space and momentum enter this picture? We give it the name of memory, and imagine that the benefit conferred by memory outweighs its cost, which is basically due to the need to control its interactions with its environment and itself in a manner that it can be easily written and read, and does not change state between writes.
By logical continuity we mean the calling of snippets of code by some controlling routine which uses the snippets of code to handle some complex task like modelling global climate for the next century given the transient input of human activity.
The whole thing is built out of basic binary interactions governed by the logic of non-contradiction.
[page 80]
All true communication is digital in the sense that it is encoded in orthogonal symbols that do not overlap and which therefore cannot be confused with one another.
MAXIMUM ENTROPY = EQUIPROBABILITY
Although mathematicians may feel that the continuum gives maximum resolution because it contains a transfinite number of points, the information theorist will be puzzled by the expression continuous points, since it is of the essence of a point that it be distinct from, rather than continuous with, its environment, in other words a point is a mark and a continuous mark, like an infinite white plane, carries no information beyond its mere existence. Algorithmic information theory takes us beyond geometric continuity into logical continuity. Chaitin
A TURING MACHINE IS A LOGICAL CONTINUUM
Every move is determined by the outcome of its predecessor.
Turing machines can be 1, 2, . . . n dimensional, as measured by the number of elements in the tape used to represent their states. (Davis Davis)
Why do continuous complex periodic functions figure so strongly in quantum mechanics? Is it because the hardware of the Universe is a structure isomorphic to the complex numbers? Symbolic (as opposed to analogue) computation is also periodic, each large expression being evaluated by the evaluation of the smaller expressions into which it might be broken.
[page 81]
Shannon's proof of his theorem about transmitting information over a noisy channel is worked out in a function space similar to the Hilbert space of quantum mechanics. Shannon
Lonergan's work is flawed by the mandatory assumption that God is not the Universe, leading to the uncritical assumption of an hypothesis about the world which is incompatible with the possibility of their identity.
A Turing machine is a universal computer, but we can imagine making a universal computer out of a network of non-universal computers. All we have to do to make a physical computer is to emulate Boolean algebra, and to enable instances of these logical operations (not, and, or) to communicate with one another so that the output of one can become input to another, all orchestrated by a stored program. An automaton, in other words. So an isolated quantum system is to be modelled as a deterministic evolution of a superposition of an infinity of frequencies between say 0 and some sort of infinity.
One hope for natural theology is that it would encourage the clerics of the world and their followers to move bit closer to reality, particularly in the recognition that the past need not and in fact cannot fully constrain the future.
Tuesday 4 August 2009
The initial singularity is an isolated quantum system, and really the only one. The coupling between isolated system must take place within the isolated system, rather than outside it, and
[page 82]
our basic question is how does a single isolated system complexify within itself to become the Universe we observe. Our persistent clue is the Trinity and Cantor's theorem.
The wave function of the initial singularity is the wave function of the Universe as that point, represented by the Schrödinger equation i d psi / dt = H psi. We might begin with H = E and then the energy begins to split up into different channels until we have something as complex as this Universe. The complexification of the differential equation. How does it complexify? By communication. The scientific act of creation described by Misner, Thorne and Wheeler is also the ontological act of creation, so the Universe is effectively embodied mind.
Then we develop the transfinite network to give mathematical substance to this rather vague idea. It all revolved around layered formalism and symmetry with respect to complexity which is preserved by normalization at each transfinite layer of complexity. Noether's theorem: what is conserved is probability / unitarity / normalization. This happens 'inside' the initial singularity. [what is not conserved is cardinal, which increases without limit while energy is conserved]
Creation of a network. A network is a connected set of messages, and the connections may be deterministic or probabilistic, ie meaningless and meaningful. Information becomes meaningful by breaking a symmetry. Revolving in an infinite dimensional space trying to find the right angle.
Isolated system develops an infinity of interacting copies of itself - the procession of the word, a perfectly good psychological model which has been 'numenized' by religion.
[page 83]
What we are seeing in the creation of a message and two nodes is the action of Cantor's theorem on the formalism of the Universe.
A differential equation (any equation) is a potential [ie a fixed point in the motion]
We are inclined to think of mass as passive, but physically it is a measure of energy that is rate of action.
Messages are created and annihilated in a network, written and erased (written over).
ERASE = WRITE OVER ==> CONSERVATION OF ACTION (ie each symbol is an act and the cardinal of the set of symbols (ie memory size) is fixed.
STATE = message in a memory, eg an electron at a certain address. Physics is interested on the mapping of addresses to 4-space.
PHYSICAL vs LOGICAL
At the level of the Schrödinger equation, physical = logical?
The halting problem: execute it and see!
Wednesday 5 August 2009
Still as ever, trying to bring this thing to a conclusion and as usual coming up against intractable details that seem difficult to wave away.
[page 84]
We start the Universe with a one dimensional Hilbert space governed by the energy equation i d psi / dt = E psi = H psi. Hilbert spaces increase their dimensionality by communication exactly like the Trinity, a 2D Hilbert space results from 2 1Ds communicating, giving a two state system. ie two separate memories. Already we are bringing in quantum field theory.
The energy equation is the fixed point in the initial singularity, matched by the dynamics so that the total energy of the singularity and all its children is zero.
Thursday 6 August 2009
From sunshine to love along the thermodynamic way, a radiant way,Cantor's theorem is implemented by communication as the initial singularity goes from 1 [0 ?] to aleph(n) dimensions.
Christie Rye page 185: 'Ah well, you're a righteous woman, Jane Marple and right must prevail.' ie consistency. Christie
Computation is a periodic process which explains the utility of complex numbers in quantum mechanics -- not so sure of the simplex / duplex explanation although it fits in somewhere. A big computation comprises a vast number of little computations right down to the unary operation of negation, which is perhaps characteristic of 1D Hilbert space, expanding to binary operations in 2D etc. Given the same software, the power of computers is a function of world length and clock frequency. The quantum of action may be associated with words of unit length and the system builds up in layers of
[page 85]
increasing world (message) length.
I am tense because I want to go faster than I can, but once I have reached my local velocity of light (taking no distractions of slowing input from the outside world) I can go no faster and so must learn to relax and follow my own slow progress to deeper and broader (conical) insight a la Lonergan. Lonergan
A particle with rest mass can communicate without losing its identity as the momentum of an electron by a magnetic field. A photon cannot do this, and the graviton sees no space but rather is a representation of the degree of entanglement between a parent Hilbert space and its children.
HILBERT SPACE = TURING MACHINE. 'rotating' in its own space.
INHIBITED ATTRACTION prevents collapse, for if the net attraction was monotonic, the electrons would fall into the nucleus so to speak. So there is a region of stationary (optimal) distance which in quantum mechanics is determined by the phase of de Broglie waves. Matter wave - Wikipedia
From bringing up children to caring for oneself to ruling our Universe, we all have to do a lot of things every day, more or less pleasurable. One can imagine that from an animal every action is pleasurable because this is what they have evolved to do and the pleasure it gives them encourages them to do it.
PLAYFELLOW vs PLAYMATE
The memory structure of the unverse is physical, and higher layers of the Universe use the memory of the lower to create 'state space' representations
[page 86]
of their doings, like being me.
The transfinite network is built on a countable alphabet of states.
We can represent a computation (the evaluation of a function) as a path through a Hilbert space, that is the choreographed rotation of the state vector. Although this is a detailed network process, our boundary conditions only determine the statistical weight (energy) of each link in the network.
A Turing machine with a single processor is a serial machine [time ordered].
Property is memory, and we can divide property into private property and public property, just as computer memory is divided into unprotected and protected.
Once we begin to see the Universe as a layered communication network we can see a role for 'Shannon's Demon' implementing the creation theorem, which demon is much more powerful than Maxwell's demon.
Shannon's demon and the creation of the world.
Napoleon: Howarth Waterloo page 197: 'The emporer's staff and the senior officers knew [the rumour of Grouchy's arrival] was a lie, and Ney was angry at the deceit. But once the lie was told, only a traitor could have told the truth.' Howarth
Friday 7 August 2009
Shannon's Demon
[page 87]
All we require is that the observed world be quantized. If it is not observed (like an isolated system) we see no quantum and we can say nothing about the unobserved system.
Shannon's demon has a computer and a memory to enable it to encode and decode messages in a manner which leads to stable complex dynamic structures (ie computers). In technological networks error correction is achieved in one or a few steps, but in real life we might imagine the resolution (complexity) of structures to increase one step at a time, acting to implement Peano's axioms or Cantor's theorem, depending on how we look at it. The self bootstrapping computer. Dawkins
One of the many styles of human power tripping is springing decisions suddenly to keep the dependents off balance.
The only constraint on codes is that they be reversible (by those in the know) and computable.
We suppose that unitary evolution has no memory and its formal constraint is the Schrödinger equation, it the Schrödinger equation stays the same forever while its solutions have transfinite variety and follow one another in a deterministic sequence, according to quantum mechanics.
Can a continuous function be deterministic? Only insofar as it can be analyzed by a Turing computable discrete function, hence the epsilon / delta arguments, where epsilon and delta are given determinate values. Si classical mathematics believes in continuous determinism, but we cannot make infinitely resolved continuous (analogue) determinate machines, there is always an error
[page 88]
but we can do perfect digital calculations as long as we stick to rational numbers.Since a computer represents reals by rationals, this is as exact as we are ever going to get.
Maxwell's Demon has a difficult task, from a classical point of view, measuring the direction of a molecule to see if it will go through the door, and measuring its momentum to decide to open the door or keep it closed. From the quantum point of view, the work seems easy (ref?).
Lewis Allegory page 11: 'What is new usually wins its way by disguising itself as the old.' (?) Lewis
Real changes in human sentiment are very rare -- there are perhaps three or four on record -- but I believe they do occur and this [courtly love] is one of them.'
Courtly love: Humility, Courtesy, Adultery and the Religion of Love.
page 13: 'Any idealization of sexual love in a society where marriage was purely utilitarian, must begin by an idealization of adultery.'
Lewis page 15: Peter Lombard: 'omnis ardentior amator propriae uxoris adultor est.' (!)
Aquinas I, II, 34, 1: 'The evil in the sexual act is neither the desire nor the pleasure, but the submergence of the rational faculty which accompanies them: and this submergence, again is not a sin, though it is an evil, a result of the Fall. (Lewis page 16). Aquinas 778
Saturday 8 August 2009
[page 89]
Raizen, Demon Raizen
The union of special relativity and quantum mechanics is quantum field theory. It has been found that this union yields a relatively small set of constraints on the Universe that are verified experimentally. The oldest and best developed field theory is quantum electrodynamics, which yields results corresponding to measurements in the parts per billion range.
A fundamental symmetry of quantum field theories is gauge invariance. A standard approach in mathematical physics is to devise a differential equation which describes the local (infinitesimal) behaviour of a system and integrate it to arrive at a global picture. Each integration introduces a constant of integration which must usually be determined by measurement. In other words, information must be added to the information in the differential equation to get a description of actual reality. There seems to be more information in the applied integral that in the differential equation. On the other hand, by summing everything together, the integral seems to remove some of the information in the original function.
Local gauge invariance.
Peskin & Schroeder Peskin & Schroeder
A reversible Universe can fall to bits at any time, so that creation requires the establishment of irreversible complexities.
Szilard on Maxwell's Demon. Szilard
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