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Notes

[Notebook: Language DB 57 ]

[Sunday 26 December 2004 - Saturday 1 January 2005]

[page 50]

Sunday 26 December 2004

Nature [Commentary?] From physics to metaphysics: we trace again the path taken by Aristotle from change in the sublunary world to the unmoved mover - ie constant potential.

[page 51]

A study of molecular biology [and creativity in general] reinforces the feeling that there is no obstacle in the Universe that cannot be overcome by suitable capital investment, that is the development of a dynamic structure (process) to bypass the obstacle. de Soto This is the technological analogue of the idea that every problem has a solution. But does it? This problem has been carefully studied in the mathematical realm. Here it is found that the processes required to solve some problems require unlimited resources in (memory) space and the number of operations (counted by time). This was proved by Turing. Turing, Davis But are Turing machines the boundary of computing? A metaphysician may be inclined to say no, that there are resources for understanding situations and working out how to deal with them available in the divine and spiritual world that are unavailable in the material world.

The Greek philosophers postulated a connection between the spiritual and material worlds. Plato, fearing chaos, postulated a non-chaotic world of organized bodies of information (ideas, forms, ordered sets) that somehow impressed themselves on the material world [and human minds] to give is the structures we see and the understanding of them we have. So pure non-constructive mathematics is sometimes called Platonic.

These ideas became central to Christian modelling of the world and remain deep in the foundations of many people's world views.

[page 52]

People have dreamt for some time that quantum theory may provide us with a new window on the relationship between matter and spirit.

We might see quantum mechanics as the most interesting result so far of the program enunciated by Galileo: . . . Galilei

Prandtl Nature 432:807 16 December 2004. Narashima

This notion that the world and mathematics are somehow images of one another took a vast leap forward in Georg Cantor.

Linguistic space / chemical space / Cantor space. Dobson

Barasch N 432:811 'So what happens when bacteria grow in a host that also covets iron (mammalian blood serum has just 10-26 M free iron), or when two microbes compete for the same source of metal. Essentially, thievery reigns. Barasch and Mori p 811

Monday 27 December 2004
Tuesday 28 December 2004

[Systems] designed to handle (rare) worst cases spend most of their time effectively not stressed at all, ie in heaven.

Metric space: Every technology is founded on a metric, eg entropy and information theory.

Wednesday 29 December 2004
Thursday 30 December 2004

Cybernetics allows us to partition the set of events into various versions of controllable and uncontrollable using the principle of requisite variety (Nyquists's theorem). Ashby, Justin Romberg

CONTROLLABLE = COUNTABLE

Like Plato, Aristotle and Newton, Cantor was a theologian looking for a space in which to model the relationship between heaven and earth. In each case the search is for invariants: things that remain the same as we change our position in spacetime. Plato began with separated forms, an early version of the idea Cantor expresses as an ordered set : 'The concept of 'ordinal type' developed here, when it is transferred in like manner to 'multiply ordered aggregates' embraces in conjunction with the concept of 'cardinal number' or 'power' everything capable of being numbered that is thinkable, and in this sense cannot be further generalized. It contains nothing arbitrary, but is a natural extension of the concept of number.' Cantor page 117

Plato was inclined to derogate the visible world, but Aristotle began a move in the opposite direction. In his Physics and Metaphysics (and other books) he developed a model for the sublunary world which he then extended to the heavens. In a nutshell, this world has the attributes of potential and action (inherent in the notion of motion) whereas the heavens are pure action. In other words, every possibility in the heavenly sphere is activated, whereas in our world, only some possibilities are activated, and the activation may change from one to another.

This diarised approach tries to capture connections (forms of words) as they surface in my mind. Ultimately this set of 'shots' is to be cut and pasted into a more coherent (better ordered) construct.

[page 54]

GEOMETRICAL PROOF - CONSTRUCTION

Although Plato, following Pythagoras, was aware of the importance of mathematics, it took Newton's introduction of calculus to mathematics to enable a static formal description of a dynamic process. For Plato and Aristotle, form did not change. The evolution of species was not possible. Like other forms, species are immutable.

Calculus shows us how to deal with the dynamic transformation of forms. Dynamics revolves around the interface between countable and uncountable, which required the 300 years between Newton and Cantor to solve.

The crowning glory of the Newtonian approach is quantum mechanics. Through the works of Cantor and Hilbert, However, Quantum mechanics addresses a much larger space than Newtonian physics.

Deighton 'Only. . . page 109: '. . . Its love alright. And like a lover, he keeps pushing the relationship to the brink of dissolution. 'Disillusion corrected Bob. Deighton

We all seek a comfortable relationship with our environment (people, animals, plants, things). We can imagine two ways of creating such comfort which we will call the developmental and the Buddhist. (active, passive) Active change the environment; passive, change yourself. We then use some sort of extreme principle to decide the

[page 55]

optimum balance.

[diagram]

Descartes alerted us to the need for clear and distinct ideas. Descartes Three hundred years later information theory has shown us how to use clearness and distinctness to overcome noise [error] in communication and showed us how to communicate clear and distinct ideas over a noisy channel. Shannon Throughout, we consider Descartes 'clear and distinct idea' to be isomorphic to Cantor's ordered set.

Plato, Aristotle, Newton, Cantor all theologians trying to discern the bond between heaven and earth.

In questions of survival, time is of the essence since continued life requires continued input.

"Their knowledge and observation of the ways of the world put them into a position of wealth and power and they held onto it [?]

Unitarity - justice - due process - conservation

In the transfinite realm, unitarity is imposed upon the network by the limited size of the natural

[page 56]

numbers with respect to the transfinite numbers. This statement may sound mystical, but the aim is to so embed the mysterious side of global in our understanding of the observable features that any pejorative connotation of 'mystery' and 'mystical' is removed from our glossary, except in the case of false mysticism, ie that not consistent with the observed world.

The global quantum network (transfinite network) is unitary and subsets of it may be (approximately) unitary. The distance of each point from a unitary condition is a product of its communication (interference) with the rest of the world.

One may understand adiabatic in the sense of unitary, that is a system effectively (or abstractly) isolated from the rest of the Universe. This is the dream of quantum computation. Although this dream cannot be achieved on the first iteration, quantum error correction allows us to move asymptotically to perfection [provided we have the necessary resources, ie spare entropy]

Functional analysis

 

One of the key ideas of a mathematical metaphysics is unitarity.

Let us then treat our input (our experience) as the output of some gigantic entity each of us calls 'my environment'

. My environment is a history, a lifeline of where I've been.

[page 57]

The animal stories communicate an abstract picture of human nature from generation to generation. The same, at the opposite end of the spectrum, as the work of normative educational institutions like the Roman Catholic Church.

On the evolution of country roads (communication channels)

'odium [theologicum]' theological hate: burning at the stake; torture, terrorism etc, ie being in a state that 'justifies' [makes probable] anonymous and indiscriminate killing.

To unify religion we must see theological hate as an error to be eliminated. We can see, on the competition model, why peoples should strive to eliminate one another; we wish to propagate the cooperation side of the model which says it is best for me to promote (rather than impede) my neighbour's welfare.

Many say that global welfare can be maximized by free trade, that is allowing any two individuals to enter into any relationship of exchange which is mutually agreed, ie a deal or a continuum of dealing.

Information ages at all timescales and it requires sampling at the Nyquist frequency to keep up to date.

Tsunami: disaster on a large scale, a big 'error', transient, and now we are in the decay period. One

[page 58]

proxy for this is the death rate, the number of people being killed per day. Initially this is very high ('prompt deaths') and then the rate decays. The more quickly it decays (due, eg to prompt aid), the smaller the integral of death (and suffering). Suffering can be reduced even more by preparedness. So, given sufficient capital, all habitations on earth are designed and built to minimize the risk from all sources, to the inhabitants. This is a consequence of the harm minimization approach to problems (at all scales)

One's 'integration interval' is one's life so far. As we get older, we integrate a large range of event.

So imagine that every point on the 'natural' line (ie row of points) is an event, and that these events may be arranged in any order, ie we think of them as abstract network protocols that each gives us insight into some concrete situation. So physics is generally a case of applying the dominant paradigm to more and more cases, always on the lookout for the case that lies outside current doctrine.

CANTOR replaced the notion of point with set. The information content of a point is a function of the dimensionality of the space in which it lies and the definition of the coordinate at the point in each spatial dimension. Cantor's theory suggests that most of the information is carried by the dimensions, since they are all orthogonal ie perfectly separated from one another like entropy free symbols.

Friday 31 December 2004
Saturday 1 January 2005

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Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics' 
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Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.' 
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Davis, Martin, Computability and Unsolvability, Dover 1982 Preface: 'This book is an introduction to the theory of computability and non-computability ususally referred to as the theory of recursive functions. The subject is concerned with the existence of purely mechanical procedures for solving problems. . . . The existence of absolutely unsolvable problems and the Gödel incompleteness theorem are among the results in the theory of computability that have philosophical significance.' 
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de Soto, Hernando, The Mystery of Capital: Why Capitalism triumps in the West andf fails everywhere else, Basic Books 2000 'The hour of capitalism's greatest triump is its hour of crisis. The fall of the Berlin Wall ended more than a century of political competition between communism and capitalism. Capitalism stands alone as the only feasible way to rationally organise a modern economy. ... As a result, with varying degrees of enthusiasm, Third World and former communist nations have balanced their budgets, cut subsidies, welcomed foreign investment, and dropped their tariff barriers. Their efforts have been repaid with bitter diappointment. ... In this book I intend to demonstrate that the major stumbling block that keeps the rest of the world from benefiting from capitalism is its inability to produce capital. ... The poor ... do have things, but they lack the process to represent their property and create capital. The have houses but not titles, crops but not deeds, businesses but not statures of incorporation. It is the unavailability of these essential representations that explains why people who have adapted every other Western invention, from paper clips to nuclear reactors, have not been able to produce sufficient capital to make their domestic captialism work. pages 1-7 
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Descartes, Rene, Rules for the direction of the mind: Discourse on the method, Encyclopaedia BritannicaB0006AU8ZG 1955  
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Galilei, Galileo, and Stillman Drake (translator), Discoveries and Opinions of Galileo: Including the Starry Messenger (1610 Letter to the Grand Duchess Christina), Doubleday Anchor 1957 Amazon: 'Although the introductory sections are a bit dated, this book contains some of the best translations available of Galileo's works in English. It includes a broad range of his theories (both those we recognize as "correct" and those in which he was "in error"). Both types indicate his creativity. The reproductions of his sketches of the moons of Jupiter (in "The Starry Messenger") are accurate enough to match to modern computer programs which show the positions of the moons for any date in history. The appendix with a chronological summary of Galileo's life is very useful in placing the readings in context.' A Reader. 
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Joachim, Howard H, and Errol E Harris, Descartes' Rules for the Directionof the Mind, Thoemmes Press: New Ed edition 1997 Product Description: Taken from the original manuscripts of Joachim's lectures on the Regulae of Descartes, this volume was reconstructed after his death from notes taken by his pupils Errol Harris and John Austin. A critical examination of the main rules for the direction of the mind and the expositions by which Descartes explains them, the work contains commentary on five main topics: the power of knowing, the nature of the intellect, Descartes's account of induction and deduction, Descartes's method of analysis and synthesis, and the notice of vera mathesis. Joachim then goes on to criticize Descartes's method and to expound his own doctrine of philosophical analysis. The last chapter offers his own concrete organic unities in opposition to the Cartesian complex natures.' Amazon 
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Papers
Barasch, Jonathan and Kiyosho Mori, "", Nature, 432, 7019, 16 December 2004, page 811-812. 'Bacteria have many ways of stealing iron from the organisms they infect. But this thievery is not one-sided, and a newly discovered device in the mammalian tool kit does a good job of keeping bacteria in check.. back
Dobson, Christopher M, "Chemical Space and Biology", Nature, 432, 7019, 16 December 2004, page 824-828. 'Chemical space - which encompasses all possible small organic molecules, including those present in biological systems - is vast. So vast in fact that so far only a tiny fraction of it has been explored. Nevertheless, these explorations have greatly enhanced our understanding of biology, and have led to the development of many of today's drugs. The discovery of new bioactive molecules, facilitated by a deeper understanding of the nature of the regions of chemical space that are relevant to biology, will advance our knowledge of biological proceses and lead to new strategies to treat disease. '. back
Narashima, Roddam, "Essay Concepts: Divide, conquer and unify", Nature, 432, 7019, 16 December 2004, page 807. 'Werner Heisenberg said that Prandtl had "the ability to see the solution of equations without going through the calculations". Prandtl demurred, "No, I strive to form in my mind a thorough picture ... the equations come only later when I believe I have understood ... [and are] good means of proving my conclusions in a way that others can accept." His papers have a simplicity and directness born of supreme self-confidence. They do not trumpet their success or criticize others, but just get on with solving the central problems using all the tools available - observation (plenty of it), mathematics, calculation and modelling. Prandtl's methodological eclecticism set the style of fluid dynamics reseach in the twentieth century. No wonder G. I. Taylor called him 'our chief' and helped nominate Prandtl for the Nobel prize he never won.'. back
Shannon, Claude E, "The mathematical theory of communication", Bell System Technical Journal, 27, , July and October, 1948, page 379-423, 623-656. 'A Note on the Edition Claude Shannon's ``A mathematical theory of communication'' was first published in two parts in the July and October 1948 editions of the Bell System Technical Journal [1]. The paper has appeared in a number of republications since: • The original 1948 version was reproduced in the collection Key Papers in the Development of Information Theory [2]. The paper also appears in Claude Elwood Shannon: Collected Papers [3]. The text of the latter is a reproduction from the Bell Telephone System Technical Publications, a series of monographs by engineers and scientists of the Bell System published in the BSTJ and elsewhere. This version has correct section numbering (the BSTJ version has two sections numbered 21), and as far as we can tell, this is the only difference from the BSTJ version. • Prefaced by Warren Weaver's introduction, ``Recent contributions to the mathematical theory of communication,'' the paper was included in The Mathematical Theory of Communication, published by the University of Illinois Press in 1949 [4]. The text in this book differs from the original mainly in the following points: • the title is changed to ``The mathematical theory of communication'' and some sections have new headings, • Appendix 4 is rewritten, • the references to unpublished material have been updated to refer to the published material. The text we present here is based on the BSTJ version with a number of corrections.. back
Turing, Alan, "On Computable Numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2, 42, 12 November 1937, page 230-265. 'The "computable" numbers maybe described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost as easy to define and investigate computable functions of an integrable variable or a real or computable variable, computable predicates and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the rewlations of the computable numbers, functions and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine'. back
Links
Alan Turing On Computable Numbers, with an application to the Entscheidungsproblem 'The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique.' back
Claude E Shannon A Mathematical Theory of Communication 'The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.' back
Justin Romberg Nyquist Theorem The Connections Project, Rice University: 'The fundamental theorem of DSP [digital signal processing]' back

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