Notes
[Notebook: Transfinite field theory]
[Sunday 1 February 2004 - Saturday 7 February 2004]
[page 40]
Sunday 1 February 2004
The problem lies in representation. We seek to capture the nature of a dynamic world in static symbols like these, F = ma, G = 8 pi T etc etc. We overcome this problem, to some extent, by using two static representations ('before' and 'after') to represent any dynamic event, and using the methods of calculus (differentiation) can sometimes take this to the limit where before and after are infinitesimally separated, and we get a static representation of the motion, the differential. The differential is local and we can sometimes make it global
[page 41]
by integration as in the general theory of relativity. Misner, Thorne and Wheeler. On the other hand, when we come to quantum mechanics, we sometimes find it necessary to use an infinite superposition of different representations to capture one state, and we imagine that all these representations point to something in the state, although when the state decays only one of the representations will be communicated to the world.
The World as Will and Representation: Schopenhauer. Schopenhauer.
Uncertainty and capital. An investment of time, energy, money etc is needed to achieve any task, but until the task is satisfactorily completed, there is uncertainty about the return on investment. We imagine that this applies across the board from the simplest particles to the largest organisations.
The fact that relativity can be perfectly represented in text (mathematical symbols) may be telling us something. It may show in Einstein's attitude to quantum mechanics: he was fixated on the idea that a good theory completely determined reality, ie was the same complexity of reality and therefore able to reduce its variety to zero, yielding certainty (p = 1) to the outcome. Ashby. We can answer the problem of quantum mechanics by saying that the complexity of quantum reality is so great that it cannot be captured in one text (measurement) and so neither we (nor the Universe?) can write a sentence that completely describes a quantum system. Instead (as in quantum electrodynamics and elsewhere) we find ourselves summing infinities. So quantum theory (any possible version that can be written with ℵ0 characters) is always incomplete, and one is surprised that it did not arise in the discussion between Einstein and Gödel that this might be the case. Dawson pp 176-177.
[page 42]
Engineering drawings and specifications strive for completeness in that they try to describe a physical structure (eg an engine) whose behaviour is one hundred per cent predictable. This idea can be approximated by tolerance, factors of safety and so on. Complete completeness cannot be guaranteed, but we can, by calculation and testing, estimate a lifetime for each thing we design.
If relativity is complete, we might say that the physical reality it applies to must be countable. Hence we can use the natural numbers as a model, and see relativity as a mathematically continuous expression of an underlying deterministic process (pregeometry) which, like the propositional calculus, is complete.
Die Welt als Wille und Vorstellung [The world as will and representation]
Making spacetime countable also makes it 'ideal' in the sense of potentially deterministic (classical, non-quantum)
[Schopenhauer] Translators Introduction.
Thesis 'On the fourfold root of the principle of sufficient reason'
'Thus the principle of sufficient reason [completeness?] deals only with our representation, in the widest sense, that is to say, with the form in which things appear to us, not with that inscrutable metaphysical entity which appears through this form, and which Kant calls the ;thing-in-itelf'. Because this 'thing-in-itself' transcends the physical framework of time, space and causality, and therefore of our cognitive functions, Kant regarded a knowledge of it as impossible. Schopenhauer admitted this up to a point, although, by identifying the Connation thing-in-itself with will in ourselves, he maintained that experience in itself as a whole was capable of explanation; yet he did not imply by this that no problem remained unsolved.'
[page 43]
Representation : ordered wet.
Preface to the first edition:
'I propose to state here how this book is to be read in order to be thoroughly understood. What is to be imparted by it is a single thought . . .
. . . a single thought, however comprehensive, must preserve the most perfect unity. If, all the same, it can be split up into parts for the purpose of being communicated, [and executed?]. then the connexion of these parts must at once be organic, ie of such a kind that every part supports the whole just as much as it is supported by the whole; a connexion in which no part is first and no part last, in which the whole gains in clearness from every part [holographic?], and even the smallest part cannot be fully understood until the whole has been first understood. But a book must have a first and last line, and to this extent will always remain very unlike an organism, however like one its contents may be. Consequently form and matter will be here in contradiction.'
Hologram : transform.
What about gravitational singularity? How do we have a singularity in a countable system? Every element of such a system is a singularity, a discrete entity coded so as to be clearly distinguishable from all its peers.
Schopenhauer models the unrepresentable ding-an-sich as will. Is that the best word in English for the lust for life that we call will. What about desire, potential, life simpliciter, will to live, etc. Wille, love, dynamism, lust, hunger etc etc Act.
A lie cannot be complete. Once the context becomes large enough, the discrepancy will begin to show. The
[page 44]
truth, on the other hand can (?) so that we tend to think that the more perfectly a text explains something the more complete it is. The most complete possible physical theory is a theory of everything.
digital = tolerant. The heart of information theory is the establishment of distinct points at a maximum distance apart in a high dimensional space. By adding dimensions (complexity, word length) we rapidly increase the size of the space so that the markers are further apart and so less likely to be confused. So why 4-space?
Monday 2 February 2004
Schopenhauer page 5: 'That which knows all things and is known by none is the subject.'
Better call it agent, for knowing is an active process, as is telling, the two ends of a communication chain. The agent is invisible (like the subject) until it acts, ie emits a message.
From the point of view of a given subject, all other entities are objects. Even my body is an object to me, so environment = {object} = representation. As Thomas would say the subject knows the object only insofar as the object is in the subject, ie the subject is correlated with the object.
'Therefore the world as representation, in which aspect alone we are here considering it, has two essential, necessary and insuperable halves. The one half , the object, whose forms are space and time and through these plurality [addressing]. But the other hand, the subject, does not lie in space and time, for it is whole and undivided in every representing being.
subject = act (process) object = text (message, static)
[page 45]
Space and time a priori in our consciousness? No. The consciousness is bound by the rules of addressing, order and communication, and space and time (gravitation) arise form these, in the way, for instance, that gravitation arises from the universal communication network.
Can we say that quantum mechanics is linear because of the no-cloning theorem? The quantum interactions of particles are total, in that each interaction involves the creation and annihilation of particles. There is in effect nothing left over, no debris, when a photon is emitted an absorbed. On the other hand, when I emit a table I must first absorb quantities of food and energy and am left, after the table is finished, with piles of sawdust and an environment of waste heat. Perhaps we are not so much talking about linearity here as dissipation.
Gravitation is non-linear because the field is a source, so that a field can feed on itself to make a black hole?
We think of energy as processing rate. Misner, Thorne and Wheeler hold that (given a certain mathematical structure) the whole of General Relativity is an expression of conservation of energy, ie conservation of processing rate. This means that the overall processing rate of the Universe stays the same form beginning to end. As energy can move around, so can processing rate, so that the action has different spatial addresses at different times. This picture should give us a handle to re-derive the equations of gravitation from the properties of a computer network. This is the dream, but how? Keep chiselling away at it!
Tuesday 3 February 2004
A clue to the relationship between the conservations of action, energy and momentum. Misner box 15.2 page 379 and sec 15.7 'From conservation of moment of rotation to Einstein's geometrodynamics, a preview'
Misner Thorne and Wheeler page 380: 'identify the stress energy tensor (up to a factor . . . ) with the moment of rotation;'
The clue is that action has the same dimensions as angular momentum.
Misner page 499: 'Action principle and dispersion relation (Hamiltonian) are rooted in the quantum principle: Feynman's principle of the democratic equality of all histories. Feynman and Hibbs.
Evolutionary tautology says the fit survive (the survivors are fit). We can model 'fitness' through minimal action. The fit entity is the one that can get what it needs with the least effort (action) relative to its capabilities, or, in a starvation context, can maintain life with the least action.
Wednesday 4 February 2004
For a long time I have been thinking of energy as a measure of processing rate (ie actions per second). How then do we understand positive (kinetic) and negative (potential) energy. First assume that the conservation of energy which we observe can easily be accounted for by the assumption that the net energy of the Universe is zero, every appearance of positive energy being accompanied by the creation of an equal amount of negative energy, that is potential energy. We think of potential energy as the energy of position or structure, static in other words. So
[page 47]
we can correlate stillness and motion with negative and positive energy. Everywhere we see a bit of stillness, we know that somewhere kinetic energy exists whose absolute value is equal to the energy of the structure observed. This text, for instance, is a static structure created by the flow of kinetic energy in my body. When if ever, it is read it may create kinetic energy in the reader. This is not a clear example. Simple harmonic motion is better, like a pendulum. Here kinetic and potential energy swap back and forth as the pendulum swings. The change in the potential energy from the bottom to the top of the swing is mediated by the gravitational field in which the pendulum moves. This is a conservative field, so the energy accounts remain arithmetically perfect. The change from human movement to text and back again is not so conservative. The field in which this transfer occurs we might call mind. Gravitation, in this analogy, is thus a feature of the universal mind.
[if the energy of the Universe is algebraically zero, then so is its mass?
Can Maxwell's demon make money on the (stochastic) share market?
Thursday 5 February 2004
Going over old notes, 1983 observing the mass of detail and the insights not all as good as they looked then. We go from the particular to the general, from the subjective to the object. At the heart of life are the tiny moment to moment details of life. Only gradually do we learn to see a bigger picture, how these details fit together and ways of dealing with them that keep the stress of life within manageable bounds. This is the task of religion, to understand and order life so as to keep everyone within the envelope of their capabilities
[page 48]
while supplying all that each needs to a 'decent' life.
This is the opposite of the Aristotelian notion [Lonergan's heuristic structure'?] that we go from the general to the particular - babies call every man father.
The question comes down to how we define general and particular. The key notion is complexity. Photons, we assume, are quite particular particles of low complexity, but they are very common, ie general. On the other hand, complex entities like ourselves are not numerically very common but from our very complexity have a very wide (general) range of powers. Why is this so unclear? The key to the whole story is that entropy grows exponentially with complexity, measured as the cardinal number of a set. If card(s) = n, the number of possible 'complexions' of S is n!, so the entropy of S is log n!, which Stirling tells us grows like nlogn (?).
It is so much easier to feel things than to know how to do them,. Further, it can often take a long time to get clear feedback on the success or failure of a particular policy or action. So we feel to some extent in the dark, but like to think our evolutionary heritage has sculpted our feelings into a consistent set spanning the whole range of activity needed for survival, and so can be trusted. It is clear, however, that feeling can also be perverted, to lead to events like the Hutu genocide of the Tutsis in Rwanda. Gourevitch.
We see a communication channel as a function (of functions) which transforms input into output. We envisage this model being complex enough to encompass, for instance, the human visual channel running from environmental photons to visual images and plans of action. We can conceive similarity of the government channel which runs from all the data contributed by citizens, industry etc through the executive levels of government who conceive images and actions derived from this data.
[page 49]
In every situation we enters a decision must be made about whether action is necessary or even possible, and what to do.
Friday 6 February 2004
It is not just the whole world that is quantized. Just communication. In the transfinite network, we associate communication, quantization and observable phenomena with the countable numbers.
Most dreamers do not appreciate the momentum of the existing system. Momentum = detail.
