vol VII: Notes
2013
Notes
[Sunday 6 January 2013 - Saturday 12 January 2013]
[Notebook: DB 74 CREATION]
[page 66]
Sunday 6 January 2013
In a sense I have gambled my life on a concept which has gradually become clearer as I go along. I have been motivated to 'blame' my entry into the Dominicans on my Catholic indoctrination, but when I remember the intellectual excitement of discovering philosophy and theology as the contexts of science. This went on as I began to explore the idea that God is not other than the Universe (How Universal . . . ), which idea in turn led to conflict with the management and then expulsion. I went to university and got a degree and developed my idea in isolation. In the nineties I went to the Australian Catholic University but left, driven once again by confessional constraints. That sent me back to isolated development (assisted by many books and articles). Now I wonder if I should have found a place where I could have found academic support, but the sacrifice of life might have been too much, as it was in the Order, where the excitement of discovery was accompanied by the pains of a very constrained way of life. In a way I have devoted my life to an empirical and theoretical exploration of why the concept that God especially loves people who follow absurd monastic ideals is wrong,. All we need to get along with God is informed prudence, a no regrets policy with respect to the divine environment (had rather swollen and writing a bit rough). Christianity has cast the world as a bad thing to be avoided.
The foundation of mathematical theology is the Church-Turing hypothesis, a representation of the fundamental
[page 67]
fixed point of the Universe described by the energy equation.
The energy equation is a dynamic implementation of Church-Turing
Monday 7 January 2013
Dauben, Cantor 1979 - 1990 Dauben
Dauben page 2: '. . . Cantor's cycles of manic depression contributed in a unique and heretofore unsuspected way to his own interpretation of transfinite set theory itself.'
page 4: '. . . the book represents a study of . . . the emergence of a new mathematical theory.' which can model its own emergence.
page 6: '. . . functional analysis . . . spurred Cantor's interest in point sets and inspired his discovery of transfinite numbers.'
page 11 Dirilichet: A function is 'any correspondence between give domains [I like this].
Deplatonization. Rolf Landauer: Information is a physical entity
Addition and multiplication may be products of the 'bifurcation' of the logical 'and'. In quantum mechanics repeated multiplications carry the complex phase through 2π, whereas repeated additions give the interaction of many phases at a given point.
Although there are ℵ0 shortcuts, there are many more things that can be achieved by brute force = 1 to 1 correspondence.
[page 68]
'Hankel's method of creating functions which were discontinuous on all rational points of a given domain of definition must have stimulated Cantor's imagination' On the way from continuous functions to permutations.
Does the program decode the message (into some user protocol) or the message decode the program, as in finding the program, given input and output [reverse engineering].Incomputable = indeterminate = symmetry = continuum
The invisibility theorem = uncontrolled self-reference is fatal to any system.
Tuesday 8 January 2013
Wednesday 9 January 2013
Thursday 10 January 2013
Roger Akroyd and logical continuity (consistency) Christie
Christie page 205: 'But Mrs Akroyd is totally incapable of pursuing a straightforward course on any subject.' Political rather than scientific.
We begin with two observations. 1: Time goes inexorably forward, there is no turning back and every moment, taken in its context, is unique. 2. The future is partially predictable and partially unpredictable. We can interpret these observations through the formalism of quantum mechanics, interpreted in turn as a description of a computer network [= activated transfinite set theory].
The principal force field driving us is the fact of death and the consequent need to reproduce, In fact the whole of
[page 69]
creation is in effect overcoming death. The Christian appeal to the fantasy that we live forever, despite the evidence to the contrary, will no longer do in this critical age. We are beginning to feel the need to get real.
Death shall be no more;Death shall die. John Donne
I feel quite strongly that I love you but how well do I know my own heart, a product of a fourteen billion year mating game.
Christie, Experimental method, Akroyd page 228-31: Reenact, make theory concrete.
We are offering an interpretation of transfinite set theory as the formal foundation of the observed world.
The structure of the mathematical network is bound together by proofs, that is the execution of deterministic computers, implementing simple [algorithms].
Dauben page 99: 'The shift in notation from ∞ to ω thus reflected a significant transformation in his thinking, specifically the advance in status of the transfinities from symbols to numbers.'
Page 103: Cantor: ' "The existence of the infinite will never again be deniable while that of the finite is nevertheless upheld. If one permits either to fall, one must do away as well with the other." '
[page 70]
Dauben page 107: 'Basis to the progress of all sceince, [Cantor] felt, was an acceptable concept of continuity.'
page 108: 'Cantor refused to tolerate any views of the continuum as an a priori categorization, a product of intuition not subject to analysis by thought, and hence by mathematics.'
page 109: 'Drawing on his theory of real numbers, Cantor anted to develop a purely arithmetical analysis of the continuum.'
Connectedness: Cantor 'Definition We call T a connected set of for any two points t and t' of T, there always exists a finite number of points t1, t2, . . . , tν of t such that the distances tt1, t1t2, . . . , tνt' are all less than any given arbitrary small number ε.
Dauben page 110: Cantor: Connectedness = everywhere-denseness.
page 126: 'In [Cantor's] own mind, the applications of transfinite numbers in physical terms was direct evidence of their very real existence.'
We might understand the transition from one transfinite number to the next by an implementation of the principle of general covariance, considering all possible mappings from model to reality.
page 129: 'Logical consistency was the touchstone which Cantor
[page 71]
applied to any new theory before declaring it existent and a legitimate part of mathematics.
Why does resolution grow? Formally Cantor's theorem, dynamically symmetry of general covariance. We use frames of reference to describe the world but we also use them to constrain the world to certain states we have chosen for their benefit to us, like the set of states that constitute a clock. Movement (trading) occurs at equipotentials (equal prices [buyer and seller]).
The existence of an infinite implies infinite resolution which may be defined by a very finely divided scale or by strings of position significant digits.
Dauben page 131: '[Cantor's] work in analysis and set theory has always assumed the reciprocal and unique correspondence between the real numbers and points on a real life.'
Consistency and death. What is the relationship which makes death necessary in a consistent Universe? The limits to control, requisite variety.
page 132: : Cantor: ' "Because of this extraordinary position which distinguishes mathematics from all the other sciences, and which provides an explanation for the relatively free and easy way of pursuing it it especially deserves the name of free mathematics, a designation which I, if I had the chance, would prefer to the now customary "pure" mathematics." '
page 140: 'Above all, [Cantor] soon discovered among theologians of the Roman Catholic Church a level of interest and encouragement that he had never found
[page 72]
among mathematicians.'
Pope Leo XIII Aeterni Patris 1879. Leo XIII
Dauben page 145: 'Any attempt to correlate God's infinity with concrete, temporal infinity suggested pantheism.'
page 146: 'not a creator but a reporter.' 'God has provided the inspiration.' A la prophecy - don't look at me, listen to God.
Friday 11 January 2013
Dauben page 157: 'permutations transformation.'
Permutations have effect when [the] operations [permuted] are non-commutative.
At its foundation, the Universe works by complex arithetic.
The geometric continuum satisfies the spatial imagination but may not satisfy the intellect, which demands logical continuity.
One cannot help feeling sorry for the US trying so hard and yet being loved less and less. Why, that is the question? On the surface the answer may appear to be lack of civility. Americans settle their differences with gins, not with fair trade.
Saturday 12 January 2013
Dauben page 166: 'In fact the method of diagonalization provided an
[page 73]
an easy means of establishing directly that the ascending sequence of powers of well defined sets has no maximum. In other word, given any set L, it is always possible to produce from the elements of L another set M which was necessarily of a higher power than L itself.'
Dauben page 73: 'The sets [Cantor] defined as whole entities were related to the Platonic ειδος or ιδεα, also termed μικον in Plato's Philebus. These were contrasted with the απειρον which Cantor took to refer to improper or merely potential but never completed infinity.'
page 171: '[Transfinite numbers] followed necessarily and absolutely from the existence of sets themselves.'
page 176: '. . . the value of transfinite exponentiation was equally indicated by the ease with which it could be applied to substantiating earlier results. . . . If earlier work, obtained without the use of transfinite numbers, could be directly and unambiguously transferred into the language of infinite sets, then the transfinite numbers seemed to justify themselves directly.'
page 181: 'The eventual emergence of a distinctive notation for the transfinite cardinal numers was surprisingly late in the history of Cantor's set theory.'
