vol VII: Notes
2012
Notes
[Sunday 30 December 2012 - Saturday 5 January 2013]
[Notebook: DB 74 CREATION]
Sunday 30 December 2012
[page 56]
Monday 31 December 2012
Hofstadter page 264: 'Typographical rules for manipulating numerals are actually arithmetic rules for operating on numbers.'
page 267 '. . . meaning is an automatic by-product of our recognition of an isomorphism.' Which may be arbitrary or conventional, like the isomorphism between 'apple' and an apple.
So far I have established, (to my own satisfaction) that it is possible that the Universe is divine (ie it does not seem to contradict the facts, as Aquinas thought). The next step us to find a proof (based on the fixed points of the divine dynamics) that the Universe must be divine.
Cantor symmetry - chunking - Hofstadter page 306, renormalization.
It is a very long shot, trying to establish an isomorphism between the collapse of the wave function and human insight. Phenomenologically both events mark the origin of a message, a fixed point corresponding to an eigenfunction or the thought process (a process in our central neural network) that led to the insight as the eigenfunction led to the eigenvalue.
The connection is through a symmetry with respect to complexity that gives us an isomorphism between computer network processes at all levels of complexity. We introduce the 'relativity of transfinity' by saing that from the point of view of the layers whose complexity is aleph(n+1) a layer below it with complexity aleph(n) can be resolved as distinct symbols.
[page 57]
Also, from aleph(n+1)'s point of view, aleph(n+2) is continuous and cannot be resolved. Insofar as I can resolve myself, I am a set of natural numbers, a locally countable number of dimensions in the relevant Hilbert space.
A feature of the symmetry is that the process of testing for computable functions is the same at every layer, variation and selection. In a sense the computable functions select themselves because they are computable they are compressed and can thus be physically represented much more economically.
The fixed points are shaped by negative feedback which is what is represented by complex exponentials whereas real exponentials represent positive feedback and eventually blow up, that is self destruct.
Tuesday 1 January 2013
Hofstadter page 412: 'Automatic chunking . . . once an operation has been defined in a procedure, it is consideres as simple as a primordial step.'
The symmetry arises because networks are indifferent to cardinality. We outline the story in the Platonic world using the gulf between the natural and real numbers and then by Landau's hypothesis narrow this down by requiring symbols to differentiate from 1 to 2 . . . N . . .. The Hilbert Oscillator oscillates between levels of resolution, differentiating and integrating (Hofstadter Chunking) up and down the scale that we initially understand from the Platonic model of recursively generated function spaces. The competition for physical resources represented
[page 58]
by the 'natural numbers' at the appropriate level) 'prunes' the transfinite space of functions down to computable functions and then by 'natural selection' established a probability structure among the functions beginning with the cosmic clock which is always executed and so has a probability of 1 to my own existence whose a priori probability is transfinitely small.
The reason is strongly guided by the feeling, so that often well established belief can serve as an impediment to reasonable behaviour, which requires scanning as much as possible of the space of error before committing to an action, look before you leap, in other words. Looking is a very complex activity, and what we see is partly a product of how we look. These ideas are slowly resolving themselves in my mind, differentiating and becoming clearer as time goes by.
The theory of everything must hold at all levels of complexity.
A computer is a network and a network is a computer, so we can see it as a feature of networks that they come to a resolution, a classification creating the elements of its space. The full space of a language is all permutations of its elements, This is narrowed down to a finite resolution by physical constraints, like those operating in quantum mechanics. The cardinal of a situation is the cardinal of the complete set of states, 0, 1, 2, . . . , aleph, . . ..
Hofstadter page 456: Church and kleene: 'There is no recursively related notation-system which gives a name to every constructive ordinal.' They are their own
[page 59]
names! Ordinal number - Wikipedia names!
I am to a large degree an artefact of my history, a response to the sequence of interactions between myself and my environment.
Hofstadter page 561: 'Church-Turing thesis tautological version: Mathematics problems can be solved only by doing mathematics.' or we might be solving them implicitly by just living. Church-Turing thesis - Wikipedia
C=T Thesis Standard version '.. . any mental process that divides numbers into two sorts can be described in the form of a general recursive function.'
How do networks have insights (collapses of wave function)?
Hofstadter page 566: Church Turing Thesis, Hardy's version: At bottom all mathematicians are isomorphic.'
page 572: 'Church-Turing Thesis, Reductionist's Version: All brain processes are derivable from a computable substrate', ie computer network is isomorphic to quantum mechanics and mental processes.
page 579: 'AI thesis: as the intelligence of machines evolves, its underlying mechanisms will gradually converge on the mechanisms underlying human intelligence.'
page 673: Creativity and randomness.
Computational power in the individual machines of the network is limited by Turing's theorem, but we place no limit on the number of instances of Turing machines operating in parallel in the
[page 60]
network.
Symmetry and symmetry breaking in the transfinite computer network. Stretch the tiny mind.
Robert W Weisberg Creativity page 14: 'The experimental view of creativity presented in this chapter has both positive and negative implications, and these will be explored throughout the rest of the book.'
Wednesday 2 January 2013
Creativity as problem solving, ie finding a path through constraints. Walas in Weisberg 4 stages: preparation (load the problem), incubation (unconscious [invisible, transparent] process), ullumination (appearance of a solution [halting process]), verification (testing solution). Clark, Grumberg & Peled: Model Checking
Weisbert page 26: Koestler's Bisociation Theory.
page 22: Gutenberg combined letter seals and wine press. Bisociation: previously unconnected ideas brought together.
Why am I reading this book? Looking for ideas that might explain how a network of semi-autonomous users may be creative, perhaps by connecting ideas [forms], as I am trying to connect myself and Weisberg.
The natural numbers are our hardware and number theory (with a little help from Goedel and Turing) show us that number theory can
[page 61]
do anything doable in a deterministic sense, and also that [it] has the limits detected by Goedel and Turing. Each layer is the software hardware of the layer above it.? Is this a good analogy. Mot layers are soft (at least those of sufficient complexity to be subject to G&T), but the lower layers are hard because they lack the complexity to be anything else. [Cantor function space is mappings of a space onto itself, ie 'consciousness']
In the evolutionary world created by Landau, any real number which wished to implement itself must gather together a sufficient collection of natural numbers to do the job. We can work in number theory because we know it is isomorphic to a a computable network of relations.
COMPUTER - SYNCHRONOUS = PHASE LOCLED
NETWORK - ASYNCHRONOUS - PHASE FREE
Lonergan - creative insight requires divine inspiration which we are happy with since our whole selves and our environment are divine. Yet he uses this idea to prove that the world is not-God because it is not-intelligent, even though all the evidence suggested that the God-Universe created Lonergan and the rest of us.
Thursday 3 January 2013
A very significant feature of quantum mechanics is linerarity (a product of addition) and periodicity, modelled by complex exponentiation [multiplication].
ei means e multiplied by itself i times.. How does this make sense, ie we have a formal system and we are seeking a consistent interpretations = isomorphisms?
[page 62]
Computable possibilities are pruned by natural selection, but this implies memory (genotype) as well as action (phenotype). So the visible evolution depends on underlying fixed points (genes).
Things looked pretty simple in the says of the [central] genetic dogma but now we are into possibly a large number of layers of control systems and epigenetics we are reminded that a recursive system can get very complex if conditions are favourable, ie new payers of complexity 'pay' for themselves by making their 'hardware' substrates.
Nillson page 31: 'A network is said to generalize when it appropriately classifies vectors not in the training set.'
Friday 4 January 2013
The phenomenological foundations for the observation of a quantized Universe are well established at all scales from quantum mechanics to the unity of the Universe itself and bolstered theoretically by fixed point theorems and mathematical communication theory.
Saturday 5 January 2013
G H Hardy A Mathematicians's Apology Hardy
After a lifetime slogging along, as it were underground, I am beginning to surface enough to get glimpses of the divine beauty of my environment, so often noted by others.
[page 63]
Hardy page 61: 'Exposition, criticism, appreciatiomn is work for second rate minds.' (?) The first rate minds build on what has gone before by exposition, criticism and appreciation!
page 63: 'I write about mathematics, because like any other mathematician who has passed sixty, I have no longer thje freshness of mind and the energy tr the patience to carry on effectively with my proper kob [to make mathematics].
page 65: '. . . why s it really worthwhile to make a serious study of mathematics. What is the proper justification of a mathematician's life? This seems an exercise in criticism or judgement, perhaps a product of Christianity, a by product of Judaism, a religion with a very judgemental (and therefore selective and perhaps meliorist) God, Yahweh.
page 67: 'I do what I do because it is the one and only thing I can do at all well. . . . I am not suggesting that this is a defence which can be made by most people, since most people can do nothing at all well. A bit arrogant, what about just living the unjustified life, as we mostly do?
page 73: 'It is quite true that most people can do nothing well.'
page 77: 'A man's first duty, a young man's at any rate, is to be ambitious. . . . the noblest ambition is that of leaving behind one something of permanent value— . . . '.
page 78: King Gilette, William Willett. King C. Gilette - Wikipedia, William Willet - Wikipedia
page 79: Motivations: intellectual curiosity, professional pride, reputation and position.
[page 64]
Hardy page 84: 'A mathematician, like a painter or poet, is a maker of patterns.'
page 85: 'Beauty is the first test: there is no permanent place in the world for ugly mathematics.' And since mathematics represents the fixed points of the Universal dynamics, we connect this to the beauty of God.'
page 99: 'The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of natural for the task. 'Primality' seems quite incidental to 'numeracy'.
page 102; '. . . al;l approximations are rational.'
page 103: significant [mathematics] = {generality, depth}.
page 123: '. . . there is no sort of agreement about the nature of mathematical reality among wither mathematicians or philosophers. . . . I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove . . . are simply our notes of our observations.''
page 135: ' "Imaginary" Universes are so much more real that thius stupidly constructed "real" one; and most of the finest products of an applied mathematician's fantasy must be rejected as soon as they are created, for the brutal but sufficient reason that they do not fit the facts.' (?) If the fantasies are formally consistent, they are facts of pure mathematics, [fixed] points in the mathematical community.
page 136: Hogben " 'without a knowledge of mathematics, the grammar of size and order, we cannot plan the national
[page 65]
society in which there will be leisure for all and poverty for none.' " Lancelot Hogben - Wikipediapage 140: 'Real mathematics has no effects on war.' You wish. So what about the theory of computation, to which you seem blind?
JBS Haldane Callinicus J B S Haldane - Wikipedia
page 143: '. . . there is one purpose at any rate which the real mathematics may serve in war. When the world is mad, a mathematician may find in mathematics an incomparable anodyne. . . . as Bertrand Russell says "one at least of our nobler impulses can best escape from the dreary exile of the actual world.
page 145: 'Alan St Aubyn A Fellow of TrinitySaint Aubyn
Fellowship implied original work.
page 147: Jordan, Course d'analyse Camille Jordan - Wikipedia
Hardy page 148: 'A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas [create when you are creative, check afterwards].
Our symmetry with respect to complexity is embodies in the Church-Turing hypothesis.
Clark et al Model checking page 13: 'A computation is an infinite sequence of states where each state is obtained from the previous state by a transition.'
We get an electron on the screen when 'influence of hole 1' and 'influence of hole 2' is true, which means quantum mechanically that the amplitudes from the two holes
[page 66]
are in phase.
