Notes
[Notebook: DB 58 Bringing god home]
[Sunday 22 January 2006 - Saturday 28 January 2006]
Sunday 22 January 2006
Monday 23 January 2006
[page 39]
Tuesday 24 January 2006
Memory makes TRAINING possible.
Does unitarity conserve computability? Pour-El and Richards. Pour-El and Richards
The uncertainty principle relates to computation of infinite series - the longer we allow for the computation of the series the more precise the result, just as the longer we observe a stationary state, the more precisely we can measure it.
In general mathematicians seem to think something can be achieved if countable processes lead to uncountable precision.
[page 40]
Wednesday 25 January 2006
Pour-El page 24: 'On recursion-theoretic practice a real number x is usually viewed as a function N --> N. Then a function of a real variable f(x) is viewed as a functional, ie as a mapping phi from functions a as above into similar functions b. Within this theory, there is a standard and well explored notion of a recursive functional, investigated by Kleene and others. . . . Suffice it to say that these notions gave the original notion of 'computable function of a real variable": the real valued function f is called computable if the corresponding functional phi is recursive.'
Recursivity allows a finite 'program' to go through a countable number of cycles to make ever closer approximations to a real number. In any order of transfinity, the subset of computable functions of of measure zero and all the other functions cannot be computed but must be represented by look-up tables. These look-up tables are not computable but must be established by trial and error, that is by evolution.
The basic elements of drama (and evolution) is two elements communicating and reading to one another.
I often think that the whole continuous and measurable function approach to nature is a mistake, but it is a point of view as hard to abandon as the reliance of physicists on fixed coordinate systems to visualize the physical world. It seems that the appearance of continuity and measure is simply a result of the probability theory law of large numbers through Born's notion that the squared absolute value of a complex vector gives the probability of an event. It seems that we have to find an interpretation of this rather weird but well attested fact via a communication/coding/network model of the world. It is just disappointing that even though I have followed this idea since 'How Universal is the Universe' that I have still not managed t
[page 41]
to tie it down to a clear and consistent and convincing exposition.
Thursday 26 January 2006
The discovery of quantum mechanics proceeded from the observation to the theory over a period of almost thirty years. The process was much helped by the existing classical models(Newtonian, Hamiltonian and Lagrangian)
COMPUTABLE = COMPRESSIBLE
In transfinite terms, there is a computable subset (cardinal ℵ0) of the reals (cardinal aleph(>0)). Each of these computable reals (which is represented by a decimal or sequence of length ℵ0) can be represented by an algorithm of finite length, so en is computable by the sequence 1 + n + n2 / 2! + . . . .
This is a feature of analysis and computability of numbers which has very little to do with the actual processes of the world except in the large number limit. So, as I have said before, physics (= statistical mechanics) is largely a matter of traffic analysis in a communication network with very little reference to the meaning of the traffic.
Do we argue from 'conservation of probability' to unitarity, or vice versa? [they are consistent with one another!]
Recursively enumerable = can be arrived at by repeatable finite operations (serial)
Non-recursively enumerable (parallel): can exist only as a (transfinite, incompressible) ordered set.
COMPUTABLE == OBSERVABLE
NON-COMPUTABLE == VIRTUAL (?)
[page 42]
We may consider cardinal measure and ordinal measure. Ordinary measure theory is concerned with scalar (cardinal) numbers like probabilities which can be scaled on a finite interval, say [0. 1]. Ordinal measure is more like a photo or drawing of a scene, where the relative placing of elements of the scene and elements of the measure are isomorphic. Language (like this) is an ordinal measure, the ordering of the words (and their individual meanings) being in some way isomorphic to the mental state that led me to write this paragraph.
von Neumann page 6: Hamiltonian H(qi, pi) i = 1, k, describes a system with k degrees of freedom, ie a network with k nodes. L(q, q') gives us the energy of this system, ie the total processing rate, which comprises a potential (structural) and kinetic (dynamic) element corresponding to each degree of freedom (source, particle).
E = mgq, E = 1/2 m q'2.
Friday 27 January 2006
[page 43]
