Notes
[Sunday 30 November 2008 - Saturday 6 December 2008]
[Notebook: DB 64 Gravitation]
[page 183]
Sunday 30 November 2008
West: Shoes: 'The great heresy of the earthly paradise still creeps across the world like a cancer . . . . This I am pledged to fight, . . .'West
How stupid can you get? This is our paradise, tight fitted by evolution
A system which denies reality is bound to be corrupt if it is to survive, ie it must survive despite its structure.
West page 91: '. . . the Church fulfills her prime mission -- the sanctification of individual human souls . . . '
What does this mean? Indoctrination to contribute to the welfare of the Church.
The stable part of earth, of us, is a compound of memories stretching back to the initial singularity
How tough can life get? The Book of Job ; Book of Job - Wikipedia The Death of Woman Wang. Spence
re Confucius (See also Jesus, Mohammed, Buddha etc): 'From the birth of mankind until now, there has never been another like our master.' Spence page 16
[page 184]
The creativity of a network is maximized when it is peerful, not hierarchic or based on power.
Sixteen moral maxims Spence page 17. Reduce to 1 moral maxim, ie one atom of social structure.
The multiplier effect of algorithms: n bits of algorithm, applied often enough, will avoid > n bits of error.
RAPE vs LOVE - the network perspective.
Monday 1 December 2008
The momentum in the stock market is often psychological, people start buying (or selling) and others follow suit, creating a positive feedback with a life of its own.
Tuesday 2 December 2008
A mathematical foundation for a stable society: not only does an increase in entropy make possible the clarification of communication so that it becomes resistant to error, but communication itself increases entropy, so establishing a virtuous circle. The mathematical foundation of stable society
Descartes notion of clear and distinct ideas was a forerunner of communication theory. Stanford Encyclopedia of Philosophy
Lawmaking, peacemaking, judging, keeping a marriage together, designing structures, all revolve around the same
[page 185]
process of clarifying ideas by dialogue in order to arrive at a clear and error free communication. This operates at every level, It not only explains why the Universe is quantized but why the stars and galaxies are distinct individuals in the night sky.
Fides quaerens intellectum [faith seeking understanding, faith = data] Lonergan describes how insight reduces masses of apparently amorphous data into clear, distinct and easily communicable packets. Lonergan He sees insight as a power of the human mind but in fact it is ubiquitous, clearly present in the formalism of quantum mechanics which governs the microscopic world and in the creation of more and more complex compounds thereafter.
All this is steps along the way to my finding a clear, distinct, ie quantized understanding of why the Universe is quantized.
The physical world upon which we are built. Boltzmann's heroic calculations showed us that we must consider every permutation of all the molecules of a gas if we are to derives the macroscopic thermodynamic properties of the world from the microscopic behaviour that underlies them. Cercignani
Wednesday 3 December 2008
The Dirac delta function appears to mediate between continuous and discrete views of the world. Dirac Its apparent artificiality points to a certain incommensurability between these two domains and suggests that the discrete, rather than the continuous, is the one to bet on. This is consistent with communication theory, since only discrete entities are observable and open to logical manipulation. The whole of continuous mathematics is expressed (communicate) in discrete symbols. Streater and Wightman page 31 sqq. Streater & Wightman
[page 186]
There is no time in formalism. From the point of view of every observer a photon travels at c and all that one can see on it (if is could be non-destructively observed) is a stopped clock of zero length. The transmitter endows it with a certain energy which is received by the receiver, modified by their relative velocity and any gravitational fields the photon may have passed through.
You don't move around in Hilbert space, you rotate, and the same may be said, in a way, for Minkowski space. From the point of view of any observer at rest in an inertial frame, other frames in relative motion appear to be rotated to that tome takes on a spatial quality and vice versa.
There is very little information in a continuum, which is why we can describe functions with infinite numbers of points in range and domain by a few symbols, like F = ma.
Natural non fit saltum? Maybe, maybe not.
Thursday 4 December 2008
Zeno exposed some of the difficulties involved in considering a continuous line as a set of points however dense. Zeno's paradoxes - WikipediaAristotle, Physics 239b15 Physicists, however, were more inclined to believe their eyes that mathematics, and so mathematicians were obliged to spend the next 2000 years developing a model of continuity that answered Zeno's objections. This task became especially important during the development of calculus [and we have all been exposed to the result, the famous epsilon delta methods of proof in analysis Limit (mathematics) - Wikipedia]. This history eventually led to Cantor's search
[page 187]
for the cardinal of the continuum, that is the number of points in a continuum, using his newly invented set theory. Georg Cantor - Wikipedia In time Cohen showed that set theory could not answer this question. Cohen In the meantime, the development of the transformation model of quantum theory led the acrobatic Dirac to his delta function and the subsequent mathematical evolution of the theory of distributions. Schwartz
Distribution is a generalization of the notion of function which makes it possible to make precise various formal mathematical manipulations common among physicists. Streater and Wightman page 31
Dirac's function became necessary if we are to say that the probability of finding a discrete particle at a single point in a continuum [is 1] - its defining feature is that although its value is everywhere zero excepts at some named point x, its integral [over any interval containing x ] is nevertheless 1. In Cantorian terms, we may think of the domain of integration (whose length we may set to 1) contains aleph(n) points, so that each point occupies a distance 1/aleph(n), and choose one point where the value of the dependent variable is aleph(n), so that the integration becomes a transfinite sum 0, . . . , 0, aleph(n) / aleph(n) (= 1), 0, . . . , 0 = 1.
When we observe physicists and mathematicians at work, or more specifically, observe their output ('the literature') when they are struggling to use the continuous formalism to model a quantized world, we see that the literature [itself] is quantized. Not only does it come in the form if discrete articles and chapters in discrete books, but when we come to a microscopic examination of this work it is written with sequences of discrete symbols which not only include the letters of the alphabet used here, but a vast number of other symbols which serve as shorthand for various sets and operations. A minimal mathematical treatise, representing the delta function,
[page 188]
might comprise an area of blank paper with a dot on it somewhere. What the mathematicians are searching for in all this is a logically consistent expression of the interface between a quantized world (a line of points) and a continuous model, an infinitely differentiable function. In a nutshell, the discussion of continuity is quantized.
This leads us to invent a new form of continuity which we will call logical continuity. The archetype of logical continuity corresponding to an everywhere infinitely differentiable function is a proof, that is a logically watertight connection between some hypothesis (say Euclid's axioms) and some conclusion (eg Pythagoras' theorem)
Dirac's delta entered the history of the world in 1930. Schwartz got the Field's medal for work on distributions in 1950. Lax, page 543. Lax
To apply the notion of logical continuity to modelling the physical world let us define a network s a logically continuous set of independent memories. This definition we use to describe the world. Each paper is a memory ibn the network, and we see that (in many cases and at least to some degree) these memories are logically connected.
Zurek Zurek
Chaitin: Algorithmic Information Theory Chaitin
[page 189]
The mathematical development of the relationship between 'distance' and 'error' (confusion) in information theory is rather complex, but of concern only to deep thinkers in the computation industry rather than average network users.
Friday 5 December 2008
From the point of view of communication theory, a continuum us a state of unbounded confusion and so is incapable of transmitting any information: it is an unmarked blank page, [entropy] The only points that convey information in continuous mathematics are singularities and fixed points, such as the eigenvectors and eigenvalues singled out by the eigenvalue equation E psi = e psi.
We define the symmetric Universe (think symmetric group) by replacing the axiom of the power set in the axiomatic proof of Cantor's theorem with an 'axiom of permutation' which encapsulates the properties of order and permutation. From a cardinal point of view any peer layers of the Cantor Universe have the same cardinal whether we generate them by subsetting or permuting. Further, permutations can be divided into subsets or cycles of smaller closed permutations. This process means that no matter what the cardinal of a permutation, we can find finite local permutations whose action nevertheless permutes the whole Universe, just by moving my pen from a to b (and moving an equivalent volume of air from b to a ) effectively permutes the whole Universe.
Every permutation can be broken down into a series of swaps and every swap permutes the whole.
Streater and Wightman: Fields were devised to overcome the problem of action
[page 190]
at a distance (and turned out to be an alternative way of describing the behaviour of particles). 'Action' is clear enough, but 'distance' depends on the metric space we inhabit. Physicists fear action at a distance in ordinary Euclidean space and in the more complex Minkowski and Gaussian spaces. Here the distance is given by a metric which is basically Pythagorean slightly modified in 'curved' spaces, which are only locally Pythagorean and which can be adjusted through a set of metric coefficients gmn. In 4 space gmn comprises sixteen functions reduced by symmetry to 10 and space-time invariance to 6 (?).
EQUATION = PROCESS (implementation of a function at every instance of its domain, ie every discrete point.
Cohen's result says that discrete objects like sets and elements tell us nothing about the 'continuum inside the discrete' just as my listening to you talk tells me almost nothing about how our minds work behind the speeches and listenings.
The difference between construction and a circus act is the difference between digital and analogue. In an analogue system perfect stability requires perfect balance, whereas in a digital system the stable point has a wide tolerances and is easily maintained.
The natural religion project is designed to capture the hippy dream in mathematics and logic.
The current task in this overall project is to map the axioms of quantum field theory into the transfinite
[page 191]
network.
Symmetric network, Symmetric network
The basic method in physics is to create a space large enough to hold the possibilities and then to narrow down possible sequences of events in this space by laws, symmetries and optimizations, hoping to arrive at something that behaves like the world we observe.
Physics currently occupies two main spaces: the four dimensional manifold of general relativity, and the function spaces (Hilbert spaces) of gravitation. We will begin with the function spaces, since they are more complex and presumably contain the space-time manifold as a subspace. Function space is a direct descendant of set theory. Here we wish to find all the productive constraints of current physics in the symmetric Universe which is as big as Hilbert space and whose operators are Turing machines.
1: quantization for error correction - Shannon
2: special relativity = communication delay - Einstein
3: Limitation of alphabet of communication to orthogonal (unconfused,
distinct) vectors = letters of the alphabet (Zurek)
4: Dirac - transformation = encoding, decoding - constraint on
transformation is normalization.
