TECHNOLOGY IN SOFTWARE DEVELOPMENT   STRUCTURAL MATHEMATICS

TECHNOLOGY IN SOFTWARE DEVELOPMENT STRUCTURAL MATHEMATICS

1. Introducción

   • The evolution of logic is intrinsically linked to the intellectual evolution of being human, since as a science of reasoning, its history represents history itself of man Logic arises from the first moment that man, when facing the nature, infers, deduces and reasons, with the intention of understanding it and taking advantage of it for its survival [1]. There are several approaches to how to interpret the evolution of logic. Poncaire divides it into five stages or revolutions, which are presented oscillating between two great topics: from rigor and formality, to creativity and chaos. The stages are identify as: Mathematical revolution, scientific revolution, formal revolution, digital revolution and the planned next logical revolution
   • Mathematical logic
     • Logistics
     • Formal Logic
     • Theoretical Logic
     • Symbolic Logic

2. Mathematics and logic

   • Mathematics and logic During the period from 600 BC to 300 BC, the principles were developed in Greece formal mathematics. This period is known as the classic period, where its The main representatives are Plato, Aristotle and Euclid. Plato introduces his ideas or abstractions; Aristotle presents the deductive and systematized reasoning and Euclides is the character that had the greatest influence on mathematicians, when establishing the method axiomatic.
   • Platón
     • Platón (427-347 BC) attempts to establish a utopian republic in Syracuse directed by philosophers, and creates the Academy in Athens, which was not just an institution philosophical, but served as a political education to the young of the aristocracy.
   • Aristóteles
     • The treatises of logic of Aristotle (384-332 a.C.), known as Organon, contain the first systematic treatment of the laws of thought in relationship with the acquisition of knowledge.
   • Euclides
     • This Alexandrian mathematician published numerous works among which highlight the famous '' Elements '', without a doubt the best-known mathematical text at long story. The '' Elements '' are divided into thirteen books and constitute one compilation of a large part of the mathematics known in Euclid's time; his great value lies in the rigorous use of the deductive method, distinguishing between principles (definitions, axioms and postulates) and theorems, which are shown of the principles. The principles of purely geometric nature in '' Elements '' they are known as postulates; three of them ensure the existence and uniqueness of the line determined by two points; the fourth, the existence of a center circle and radio dice; and the fifth gives conditions that ensure that two lines are cut in a point.

3. Mathematical science

   • René Descartes:
     • The starting point of this French philosopher and mathematician (1596- 1650) is the universal doubt, which consists of dispensing with any knowledge prior that is not confirmed by the evidence with which the spirit.
   • Isaac Newton
     • Isaac Newton (1642-1727) is owed the discovery of the universal gravitation, the development of infinitesimal and important calculus discoveries on optics, as well as the laws that govern classical mechanics
   • Gottfried W. Leibniz
     • German philosopher and mathematician (1646-1716); he founded the Academy of Sciences of Berlin (1700). In '' Discourse on combinatorial art '' states the the need for a rigorous, exact and universal language (a purely formal language). As a mathematician, his main work (published in 1684) is the memory entitled '' New method for determining maximums and minimums '', in which exposes the fundamental ideas of the infinitesimal calculus, anticipating a few years Newton.

4. Formalization of Mathematics

   • This stage is characterized by the resurgence of the rigorous formalization of mathematics, which in the classical Greek stage was representative. The use of infinitesimals was one of the most notorious practices in the Renaissance period, for which no justification was offered. The rigorous analysis came with the elimination of infinitesimals and the presence of limits as an argument. In this period, the symbolic logic, formal school, Boolean logic, propositional calculus, induction mathematics, the calculation of sequels.
   • Guiseppe Peano
     • The enunciation of the principles of the Italian Guiseppe Peano (1858- 1932) about mathematical logic and its practical application were contained in its work '' Formulaire de mathematiques ''. Peano's axioms allow to define the set of natural numbers
   • David Hilbert
     • The German mathematician David Hilbert (1862-1943) was a bitter defender of axiomatics as the main focus of scientific problems, this is, from a closed and unmovable set of premises to build the base fundamental of any study. From the Greek sources of Euclid, he published in 1899 his work '' Fundamentals of Geometry '', in which through an exhaustive analysis and refinement of Euclidean ideas, formulated its principles of axiomatization. According to his theories, it is necessary to establish a set of postulates basic concepts before posing in a more detailed way any type of physical problem or mathematical.
   • Friedrich G. Frege
     • Together with Boole and Peano, the mathematician and logician Friedrich G. Frege (1848-1925) initiated the current of thought that, starting from the analysis of foundations of mathematics, carried out the most profound renovation and development of classical logic. He was the first to introduce quantifiers or operators and in elaborate a theory of quantification
   • .George Boole:
     • The logical and mathematical George Boole (1815-1864) applied the calculation mathematical logic, founding the algebra of logic, which in a certain way performs the Leibniz's dream of a '' characteristica universalis '' or calculation of reasoning. He The use of appropriate symbols and operating rules allows the representation of concepts, ideas and reasonings through variables and relationships (equations) between them. Boole gave a general method to formalize deductive inference, representing complicated reasonings through simple systems of equations.
   • Augustus De Morgan
     • The greatest contribution of Augustus De Morgan (1806-1871) in the study of logic includes the formulation of Morgan's laws and his work bases the theory of the development of relationships and symbolic mathematics modern or mathematical logic. De Morgan made his biggest contribution as logic reformer
   • .Georg F. Cantor
     • The German mathematician Georg F. Cantor (1845-1918) owes the idea of the '' infinite continuum '', that is, the possibility of considering infinite sets given simultaneously. He is considered the creator of the theory of irrational numbers and of the sets
   • Gentzen
     • The German Gentzen (1909-1945) formulated the proof of the consistency of a system of classical arithmetic, in which the non-elementary method is an extension of mathematical induction from a sequence of natural numbers to a certain segment of transfinite ordinal numbers
   • Bertrand Russell
     •  Bertrand Russell (1872-1970) es uno de uno de los creadores de la logística y uno de los pensadores de mayor influencia en la filosofía científica contemporánea. Lo fundamental de su obra está en sus aportes a la lógica. Decididamente antiaristotélico, llegó a afirmar que quien quería iniciarse en la lógica debía comenzar por no estudiar la lógica de Aristóteles.
   • .Kurt Gödel

5. The Digital Revolution

   • This revolution started with the invention of the digital computer and universal access to high-speed networks. Turing unites logic and computing before any computer was invented. Weiner founds the science of cybernetics. At school modern computing are present logical that have allowed advances important: Hoare presents an axiomatic system of programming systems and Dijkstra a system of verification and deduction of programs from specifications 6.1.Alan Turing: Mathematical and logical who was a pioneer in the theory of computing
   • Alan Turing
     • Mathematical and logical who was a pioneer in the theory of computing and contributed in important logical analyzes of the computational processes. The specifications for the abstract computer he devised (called the Machine of Turing) turned out to be one of his most important contributions to the theory of computing. Turing also proved that it is possible to build a universal machine that with proper programming you can do the work of any machine designed to solve specific problems [9] .Alan Turing invented the machine that takes its name (Turing Machine) in an attempt to determine if all the Mathematics could be reduced to some simple type of computation.
   • Norbert Weiner
     • The North American scientist Norbert Weiner (1894-1964) in 1947 publishes his most famous book: '' Cybernetics, or control and communication in the animal and machine''; where the word Cybernetics is used for the first time. There are many Definitions of Cybernetics (from the Greek kybernetes, pilot), Norbert Weiner gave life to the word by means of a very simple definition: '' Science that studies the translation of biological processes to machine processes ''. In the beginning, Cybernetics was very linked to sciences such as neurology, biology, robotics and artificial intelligence
   • Alfred Tarski
     • Mathematical and logical Polish born in 1902, who made important Algebra studies in general, measurement theory, mathematical logic, theory of sets, and metamathematics

6. The next logical revolution

   • The next logical revolution will be the practical assimilation of mathematics and computing within logic. Emphasis will be placed on computers exploiting the information intelligently, moving from databases to knowledge bases

7. Conclusión

   • The logic shows a very interesting historical evolution, born of the strong formalization of the mathematics of the Greeks, which was impacted, as many sciences, by the thought of the Middle Ages, where religion came before everything; but, the impetus of the mind of the Renaissance philosophers helped to resume their development. There is no doubt that logic has a fundamental impact, as a science of science, in the contemporary thinking, and that the birth of computer technology should much to the development of the logical formalism of the beginning of the century