vol 2: Synopsis
part IV: Divine dynamics
page 21: Alan Turing
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... to restore theology to the mainstream of science
Alan Turing
(1912-1954)
What Goedel 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 computable problems which can be solved, but there are many other incomputable problems which cannot. Goedel's result that 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.
Turing's place in history is guaranteed by his paper 'On computable numbers, with an application to the Entscheidungsproblem'. Davis 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.
Turing modelled the 'definite method' of Hilbert's problem with an abstract computer now called a Turing machine. In Turing's day, the term computer meant a person who performed computations. Turing imagined a machine that could do everything a computer could do, read, write and erase symbols, and subject symbols to certain arithmetical and logical operations like addition, multiplication, subtraction, division, 'anding' 'oring' and so on.
A universal Turing machine can be programmed to perform any computable task. Armed with this machine, Turing then proceeded to demonstrate that there were assertions which it could not decide. Turing concentrated on 'computable numbers' which he described as 'the real numbers whose expression as a decimal is calculable by finite means'. The essence of the decision problem could be captured by dealing with the computable numbers, and then extended to other fields of mathematics.
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 aleph(0)), whereas the class of real numbers is non denumerable (its cardinal number is aleph(>0). This means that the computable portion of the real numbers is infinitesimally small compared to the total class of real numbers.
The Turing machine is a completely determinate machine. It never comes to a point where it must seek outside help to arrive at a decision. Turing imagined another class of machines whose motion is only partially determined by their internal configurations. Such machines, on coming to a certain point seek information from their environment before going on. Turing calls these 'oracle' machines (as opposed to 'automatic' machines), since they consult their oracle when they need information. We will call them network machines. For each machine in a network, the rest of the network (including its user) is its oracle.
Network machines may be considered more powerful that automatic machines, since they add the power of their oracle to their own power. Nevertheless, given the infinite recursive expansion of the transfinite numbers, it would seem that there are always problems too big for any network of Turing machines, so that certain problems remain undecidable. If we model the universe as a network of Turing machines, we are led to the conclusion that the future is not completely predictable and that there is no natural end to universal process.
Books
| Beal, 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." http://www.amazon.com/exec/obidos/ASIN/0852742622/tnrp">Amazon back |
| 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.' Amazon back |
| 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.' Amazon back |
| 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 ives, how they transform our perceptionof reality.' Amazon back |
| Davis, Martin, The Undecidable : Basic Papers on Problems Propositions Unsolvable Problems and Computable Functions, Raven Press 1965 Description: '[Includes] ... the basic papers of Goedel, 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. Amazon back |
| 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.' Amazon back |
| 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) Amazon back |
| 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?' Amazon back |
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