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volume II: Synopsis

section IV: Divine Dynamics

page 27: Alan Turing

(1912-1954)

What Gödel did for the 'static' structure of the Cantor Universe, Turing did for its dynamics. He showed that, given a reasonable definition of a computer, there are some computation problems which can be solved, but there are many other 'incomputable' problems which cannot. Gödel's result shows that there will always be incompleteness, no matter how big the system. We can interpret Turing's result to mean that there will always be incomputable problems, no matter how big the computer. From this we conclude that if the whole Universe may be modelled as a computer, there are some computations it will never complete. The Universe is in this sense eternal. Kurt Goedel I

Turing's place in history is guaranteed by his paper 'On computable numbers, with an application to the Entscheidungsproblem'. The Entscheidungsproblem (German 'decision problem') was posed by Hilbert in 1928: did there exist a definite method by which any mathematical assertion could be classified as true or false? Hilbert himself felt that there were no undecidable assertions. Alan Turing, Davis

Turing modelled the 'definite method' of Hilbert's problem with an abstract computer now called a Turing machine. Turing was able to show that there are definable numbers which are not computable, thus showing that Hilbert's expectation was wrong. Further, Turing noted that the class of computable numbers is denumerable (ie its cardinal number is ℵ0), whereas the class of real numbers is non denumerable (its cardinal number is ℵ>0). This means that the computable portion of the real numbers is infinitesimally small compared to the total class of real numbers.

Our model of the Universe comprises an address space modelled by the transfinite numbers with operations on this space modelled by computers. The question then arises, how can computers which can only compute a countable infinity of different functions deal with an address space whose higher cardinals are uncountably large? Transfinite numbers - Wikipedia

The solution to this problem lies in the concept of correspondence, in different contexts called mapping, naming or, in the physical world, binding or bonding. The first step in the workings of bureaucracies of all sorts is to establish the identities of their clients so that they can be certain that all the paperwork with the same name on it applies to the same individual. Personally identifiable information - Wikipedia

We assume that a human person is a transfinite object, but we know that the bureaucratic manipulation of that person is effectively computable, since the bureaucratic process of drawing files (elements of memory), updating them and returning them to their position in an ordered filing system is quite definite and finite. By handling the file, the bureaucrat is effectively handling the person, in a very limited way, of course.

We are imagining a dynamic universe that begins as a totally simple point analogous to the classical God documented by Aquinas. By mapping onto itself (analogous to the procession of the Word in the Christian doctrine of the Trinity), this Universe generates stationary fixed points which can be observed and documented scientifically. Trinity - Wikipedia, Aquinas 161, Brouwer fixed point theorem - Wikipedia

Turing's machine is a relatively complex device, having a finite number of internal states, being able to write and erase a 'tape' of memory squares, and to move itself along the tape in two directions. A maximal Turing machine requires a countably infinite number of memory locations.

Like Gödel's theorem, Turings theorem is a fixed point in the mathematical community. Although it is a model of a universal computer, the Turing machine is very inefficient, since it spends many operations stepping up and down its tape to reach the data it needs to perform its operations. Modern computers, with random access memory, are far more efficient: they use fewer processing operations to complete the same algorithm. Because of their enormous utility a large community of programmers has arisen to increase the efficiency of computers. The discovery of new algorithms can often yield an enormous benefit. Asymptotically optimal algorithm - Wikipedia

Hamilton's variational principle can be used to derive classical mechanics (including relativity), quantum mechanics and quantum field theory. The essence of this principal is the minimization of action which is modelled as the time integral of the Lagrangian. If we consider each operation of a computer as a quantum of action, we can guess that the optimization of computing algorithms and the minimization of action are closely related. Hamilton's principle - Wikipedia

In a universe where energy is conserved (ie the rate of action is constant) we have the necessary conditions for an evolutionary process: competition for the resource of energy or computing power by a population some of whose variations can compute more effectively than others.

Computer codes are fixed points in the software community, being created and annihilated in that community. We see this process as analogous to the evolution of genotypes in the biological community, and imagine an analogous evolution of the computable functions that quantum theory uses to explain the operation of the world. Computer programming - Wikipedia

(revised 25 May 2013)

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Further reading

Books

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Beale, R, and T Jackson, Neural Computing: An Introduction, Adam Hilger 1991 Jacket: '... starts from basics and goes on to cover all the most important approaches to the subject. ... The capabilities, advantages and disadvantages of each model are discussed as are possible applications of each. The relationship of the models developed to the brain and its functions are also explored.' 
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Church, Alonzo, Introduction to Mathematical Logic, Princeton UP 1996 Jacket: 'One of the pioneers of mathematical logic in the twentieth century was Alobzo Church, He introduced such concepts as the lambda calculus, now an essential tool of computer science, and was the founder of the Journal of Symbolic Logic. In Introduction to Mathematical Logic, Church presents a masterful overview of the subject - one which should be read by every researcher and student of 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 Goedel incompleteness theorem are among the results in the theory of computability that have philosophical significance.' 
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Davis, Phillip J, and Reuben Hersh, Descartes Dream: The World According to Mathematics, Penguin 1988 Preface: 'We are concerned with the impact mathematics makes when it is applied to the world that lies outside mathematics itself; when it is used in relation to the world of nature or of human activities. This is sometimes called applied mathematics. This activity has now become so extensive that we speak of the "mathematisation of the world." We want to know the conditions of civilisation that bring it about. We want to know when these applications are effective, when they are ineffective, when beneficial, dangerous or irrelevant. We want to know how they constrain our lives, how they transform our perception of reality.' 
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Davis, Martin, The Undecidable : Basic Papers on Problems Propositions Unsolvable Problems and Computable Functions, Raven Press 1965 Description: '[Includes] ... the basic papers of Gödel, Church, Turing, and Post in which the class of recursive functions was singled out and seen to be just the class of functions that can be computed by terminating processes. Also presented is the work of Church, Turing, and Post in which problems from the theory of abstract computing machines, from mathematical logic, and finally from algebra are shown to be unsolvable in the sense that there is no terminating process for dealing with them. Finally, the book presents the work of Kleene and of Post initiating the classification theory of unsolvable problems. Already the standard reference work on the subject, The Undecidable is also ideally suited as a text or supplementary text for courses in logic, philosophy, and foundations of mathematics.'  
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Frege, Gottlob, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Nature of Number, Northwestern UP 1980 Jacket: 'The book represents the first philosophically sound discussion of the concept of number in Western civilisation. It influenced profoundly developments in the philosophy of mathematics, general ontology and mathematics.' 
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Hodges, Andrew, Alan Turing: The Enigma, Burnett 1983 Author's note: '... modern papers often employ the usage turing machine. Sinking without a capital letter into the collective mathematical consciousness (as with the abelian group, or the riemannian manifold) is probably the best that science can offer in the way of canonisation.' (530) 
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Weizenbaum, Joseph, Computer Power and Human Reason: from Judgement to Calculation, W H Freeman 1976 Jacket: '[This book] has fired enormous controversy and acclaim in America. Here JW, one of the world's top computer scientists, provides is with an insider's critique of computers: what they can already do, what they cannot do and, most controversially, what they should not be used to do. Should we, for example, be working toward the use of computers as substitutes for doctors or psychotherapists?' 
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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
Aquinas 161 Whether any procession in God can be called generation? 'I answer that, The procession of the Word in God is called generation. . . . the procession of the Word in God is generation; for He proceeds by way of intelligible action, which is a vital operation:--from a conjoined principle (as above described):--by way of similitude, inasmuch as the concept of the intellect is a likeness of the object conceived:--and exists in the same nature, because in God the act of understanding and His existence are the same, as shown above (14, 4). Hence the procession of the Word in God is called generation; and the Word Himself proceeding is called the Son.' back
Asymptotically optimal algorithm - Wikipedia Asymptotically optimal algorithm - Wikipedia, the free encyclopedia 'In computer science, an algorithm is said to be asymptotically optimal if, roughly speaking, for large inputs it performs at worst a constant factor (independent of the input size) worse than the best possible algorithm. It is a term commonly encountered in computer science research as a result of widespread use of big-O notation.' back
Brouwer fixed point theorem - Wikipedia Brouwer fixed point theorem - Wikipedia, the free encyclopedia 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself. back
Computer programming - Wikipedia Computer programming - Wikipedia, the free encyclopedia 'Computer programming (often shortened to programming or coding) is the process of designing, writing, testing, debugging, and maintaining the source code of computer programs. This source code is written in one or more programming languages. The purpose of programming is to create a program that performs specific operations or exhibits a certain desired behavior. The process of writing source code often requires expertise in many different subjects, including knowledge of the application domain, specialized algorithms and formal logic.' back
Eigenfunction - Wikipedia Eigenfunction - Wikipedia, the free encyclopedia 'In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has Af = λf for some scalar, λ, the corresponding eigenvalue.' back
Hamilton's principle - Wikipedia Hamilton's principle - Wikipedia, the free encyclopedia 'IIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action (see that article for historical formulations). It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and has even been extended to quantum mechanics, quantum field theory and criticality theories.' back
Kurt Goedel I On formally undecidable propositions of Principia Mathematica and related systems I '1 Introduction The development of mathematics towards greater exactness has, as is well-known, lead to formalization of large areas of it such that you can carry out proofs by following a few mechanical rules. The most comprehensive current formal systems are the system of Principia Mathematica (PM) on the one hand, the Zermelo-Fraenkelian axiom-system of set theory on the other hand. These two systems are so far developed that you can formalize in them all proof methods that are currently in use in mathematics, i.e. you can reduce these proof methods to a few axioms and deduction rules. Therefore, the conclusion seems plausible that these deduction rules are sufficient to decide all mathematical questions expressible in those systems. We will show that this is not true, but that there are even relatively easy problem in the theory of ordinary whole numbers that can not be decided from the axioms. This is not due to the nature of these systems, but it is true for a very wide class of formal systems, which in particular includes all those that you get by adding a finite number of axioms to the above mentioned systems, provided the additional axioms don’t make false theorems provable.' back
Personally identifiable information - Wikipedia Personally identifiable information - Wikipedia, the free encyclopedia 'Personally Identifiable Information (PII), as used in information security, is information that can be used to uniquely identify, contact, or locate a single person or can be used with other sources to uniquely identify a single individual. The abbreviation PII is widely accepted, but the phrase it abbreviates has four common variants based on personal, personally, identifiable, and identifying. Not all are equivalent, and for legal purposes the effective definitions vary depending on the jurisdiction and the purposes for which the term is being used. Although the concept of PII is ancient, it has become much more important as information technology and the Internet have made it easier to collect PII, leading to a profitable market in collecting and reselling PII. PII can also be exploited by criminals to stalk or steal the identity of a person, or to plan a person's murder or robbery, among other crimes. As a response to these threats, many website privacy policies specifically address the collection of PII, and lawmakers have enacted a series of legislation to limit the distribution and accessibility of PII.' back
Transfinite numbers - Wikipedia Transfinite numbers - Wikipedia, the free encyclopedia 'Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary workers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use.' back
Trinity - Wikipedia Trinity - Wikipedia, the free encyclopedia 'The Christian doctrine of the Trinity defines God as three divine persons (Greek: ὑποστάσεις) the Father, the Son, and the Holy Spirit. The three persons are distinct yet coexist in unity, and are co-equal, co-eternal and consubstantial (Greek: ὁμοούσιοι). Put another way, the three persons of the Trinity are of one being (Greek: οὐσία). The Trinity is considered to be a mystery of Christian faith' back

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