THE HISTORY OF MATHEMATICS

1. Mathematical science After declining the classical school of the Greeks, a period is presented in which religious authority brutalized intellectual creativity. The Renaissance begins a new era in which the revitalization of science and mathematics is allowed. The most outstanding representatives of this stage are Descartes, Newton and Leibniz. This period covers from 1500 to 1800 4.1.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. Descartes doubted all received teaching, all acquired knowledge, the testimony of the senses and even the truths of rational order. At this point, he finds a truth that he can not doubt: the internal evidence that manifests itself in his own subject (I think, therefore I am). As a scientist, it is due to Descartes, among other contributions of considerable importance, the creation of analytical geometry.

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

3. conclusion 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 contemporary thought, and that the birth of computational technology owes much to the development of logical formalism at the beginning of the century.

4. Introduction 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

5. 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. Very remarkable characters of this stage are: Peano, Hilbert, Frege, Boole, Morgan, Gentzen, Russell, Gödel and Whitehead. A Rusell and Gödel must the approaches of the limitations of logic and science in general.

6. 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 that have important advances: Hoare presents an axiomatic system of programming systems and Dijkstra a system of verification and deduction of programs from 6.1.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 I have 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 has the proper performance and 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. Your objective was to develop the simplest possible capable computing machine.

7. Mathematics and logic During the period from 600 BC to 300 BC, the formal principles of mathematics were developed in Greece. This period is known as the classical period, where its main representatives are Plato, Aristotle and Euclid. Dish introduces his ideas or abstractions; Aristotle presents the deductive and systematized reasoning and Euclid is the character that had the greatest influence on mathematicians, when establishing the axiomatic method. In '' Elements '', Euclides organizes deductive tests within a systematic, rigorous and well organized presentation of mathematical knowledge. 3.1.Platón: Plato (427-347 BC) tries 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. According to many critics, Plate builds his theory of knowledge to justify the power preeminent of the philosopher and part of the Socratic thoughts: the search for stable concepts and definitions of abstract ideas (justice, goodness, value, etc.).

8. Summary A brief historical description of the logic and its possible future development is presented. This description is a summary of several electronic lines, where the information presented by Henri Poncaire stands out. This historical description appears ordered in the five revolutions indicated by Poncaire, and within each one it reviews its main representatives

9. Mathematical Logic The objective of mathematical logic is to question with the greatest rigor the concepts and the rules of deduction used in mathematics, constituting the logic for that reason a true metamathematical. A mathematical theory considers defined objects (integers, for example) and defines laws that relate these objects to each other (the axioms of the theory). From the axioms new propositions (theorems), and sometimes, new objects are deduced. The construction of formal systems (formalization), cornerstone of mathematical logic, allows to eliminate the arbitrariness in the choice of axioms and to define explicitly and exhaustively the rules of mathematical deduction