About Turing: Basic science versus applied science

Few problems have a more theoretical character than determining if there are more sets of natural numbers than natural numbers, if an axiomatization of predicate logic captures all inferences that are deductively valid, or if there may be a procedure that demonstrates that an inference is valid or do not. These problems do not seem to have much to do with more profane topics such as sending text messages, buying tickets for a concert on the internet, or playing World of Warcraft (WoW). However, these and other practical developments that have a profound influence on our daily lives are intimately linked to those theoretical problems mentioned initially.

David HilbertThe link is found in the computers developed from the decade of the 40s, following a design that - in its fundamental lines - had been outlined years ago by the mathematical logic Alan Turing. This is the famous "Turing machine", which was not designed to do anything practical, although later Turing saw the possibilities that such a "machine" opened for the study of intelligent behavior. The "Turing machine", on paper, was developed to solve an eminently theoretical problem: the problem of finding a procedure to determine the validity of an inference in predicate logic. The problem had been posed by a famous mathematician, David Hilbert, in 1928. The Universal ComputerBoth the formulation of the problem by Hilbert, and its resolution by Turing, are based on ideas developed by Hilbert and other logicians and mathematicians, among which include Boole, Cantor, Frege, Russell, and Godel; the "dream team" of modern logic. The story of this development is magnificently told in the book "The Universal Computer: The Road from Leibniz to Turing", Martin Davis [1].

Turing rigorously demonstrates that the procedure sought by Hilbert does not exist. Turing invents "his" machine as a formalization of what is a procedure (a method of computation), and demonstrates then that there can not be such a machine for the Hilbert problem (the Alan Turingso-called "Entscheidungsproblem" [2], to differentiate it from other problems fundamentals also formulated by Hilbert). For the test uses the idea of??coding machines with numbers, following the example of Godel who introduces a similar idea to demonstrate the incompleteness of formal arithmetic (another problem raised by Hilbert), and the technique of diagonalization, introduced before by Cantor to demonstrate that there are more sets of naturals than natural numbers, also used by Godel in his proof. See special purpose machine design

Turing does not stay in that result however and builds a "Turing machine" (still on paper) that is "universal" in the sense that this particular "hardware" can execute any procedure. More precisely, this universal Turing machine is programmable: it accepts the description of a particular machine that solves a given problem on an input, and emulates the behavior of that machine on that input. Turing machineThe universal Turing machine is the mother of the modern programmable computer. That something like that was even possible was not obvious in the 1940s when "computers" were developed to solve a specific set of calculations. The limit of the universal Turing machine, on the other hand, is the limit of current computers, the limit of the computable.

It is surprising to see how a development that was not motivated in any way by "practical" applications, has had an impact that may one day be compared to the development of agriculture. History, however, is not unusual in the development of science, where one knows where it begins but not where it ends. That is the promise of science from the social point of view, and the adventure of the scientist from the personal point of view.

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