As the crisis of the fundamentals resulted in the birth of modern computing
Computing (or computer science) has extraordinously changed the development of technology today;computers, smartphones, transmission and communication devices, including medical devices and means of transport, all these have been computing products. Information management has played a crucial role in humanity, and that is why the history of computing has been very extensive, difficult and has had many key protagonists in its development.
Mathematics are the basis of computing, they are the language on which we rely on to build, to calculate and to solve the problems. The history of mathematics is too long, but one of its many crucial moments known as "the crisis of the foundations of mathematics" and its main protagonists (such as Kurt Gödel and Alan Turing) are responsible for the birth of modern computing.
In the nineteenth century, in the rise of the industrial revolution, due to their relationship with engineering, mathematics gradually became a fundamental tool in the development of society, it is in this period of time where mathematicians were committedin improving existing concepts, theorems and theories and creating new. In 1874 Georg Cantor when raising his theory of the sets gave way to paradoxes that would form what we know today as "the crisis of the fundamentals", a period of time in which mathematicians worked hardly against each other to rethink the foundations ofmathematics and demonstrate whether or not they were a perfect science.
Mathematics used to be seen as a pyramid, or saying it in another way that they are based or simplified in simpler mathematical concepts, but what was at the bottom of the pyramid?, What was the main and most basic foundation of mathematics? These questions were those that began the crisis of the foundations, a crisis that would divide mathematicians into 3 main philosophical currents to define the foundations of mathematics (logicism, formalism and intuitionism).
The first of these, logicism, (being its main protagonists Gottlob Frege and Bertrand Russell), tried to raise logic as a fundamental basis of mathematics. "Frege was the first to argue that mathematics is simply a part of logic and, therefore, is likely to be built with pure logical procedures. Between 1879 and 1903 Frege dedicates tesoneros efforts to sitting mathematics on logical bases exclusively, the results of which he exposes in his fundamental work Grundgecetze der Arithmetik. In this work Frege makes frequent use of the notion of all sets, which leads him to a complete fiasco in his purposes, as the author himself has the courage to recognize at the end of the second volume, when he says: «A scientistYou cannot find anything less desirable than finding that the whole foundation of your work falls precisely at the time it ends.. Logicism failed to succeed because just before Frege published his work Fundamentals of Arithmetic (1884) he received a letter from Russell that presented him with the well -known “Russell paradox” that showed that what was raised by Frege and Cantor was incorrect.
Intuitionism, founded by L. AND. J. Bruwer, raised intuition as a fundamental basis of mathematics, or saying it in another way, mathematics were a construction of the human mind."Intuitionism can be understood as a particular way of incorporating the idea of constructivism in mathematics, an approach that we owe to the Dutch mathematician Bruwer and her disciple Heyting. Constructivism aims that mathematical objects exist to the extent that they have been built and that the validity of demonstrations emanates from its construction: in particular, existential statements should rely on the effective construction of their objects. Mathematical truths are created, they are not discovered. Intuitionism really never succeeded because a model that gave complete explanation to all mathematical concepts was not reached since there were many ideas that escaped intuition.
In formalism, represented by David Hilbert, it was sought to create a formal or organized system that demonstrated that mathematics were complete (which could be demonstrated), consistent (there were no contradictions) and finite (they could be demonstrated with a sequence of logical instructions).
It was called "Hilbert program" to the mission of demonstrating the veracity of formalism, to achieve this many congresses between mathematicians were made for a long period of time. When the mathematicians were getting closer to completing the Hilbert program, in one of those congresses held, Kurt Gödel demonstrated with his "incompleteness theorem" that it was impossible for a mathematical system to meet at the same time all the conditions asHe raised formalism. Gödel marked a historical point by publishing his book "On formally undecidable propositions of mathematical principle and related systems", since with his theorem he fell ridiculous everything raised by formalism, the way in which Gödel came to his demonstrations was a great greatInspiration for many mathematicians, being one of them Alan Turing.
Alan Turing, well known for his contribution in World War II deciphering the codes of the Nazi “enigma”, and also for his tragic and premature death due to harassment and rejection he received for his sexuality, Turing could be considered as the father ofThe computer science and responsible for much of the technology that we use day by day. Turing, was the bridge that connected all the new concepts obtained in the crisis of the fundamentals with the origin of computing, since Turing years before World War II inspired by Gödel, took the problem “EntscheidungSproblem” (or decision -making problem) that Hilbert raised years ago and creating his well -known “Turing Machine” showed that it is impossible to solve this problem. The "EntscheidungSproblem" raises (in simpler words) the existence of an algorithm capable of saying whether the solution to a problem exists or not;Turing, based on the concepts raised by Hilbert, Gödel and Alonzo Church, made use of his machine and wrote in his article called “On computable numbers, with an application to the EntscheidungSproblem” the following conclusions:
"1-The calculation of first-order predicates is not decided:" No machine can decide whether or not a formula theorem of the predicate calculation ".
2-There is problems that ‘no’ are computable;Thus appears the ‘stop problem’: “’not’ it is possibleN1,…, NK) will stop or continue processing indefinitely. Turing showed that there are problems that cannot be computed and therefore we do not know if there is a solution or not. Also in his article, Turing laid the theoretical basis of algorithms, memory storage, computing and even the concept of artificial intelligence.
The Turing machine is a theoretical system consisting of a tape full of boxes that can be moved from left to right, and a head that reads the tape and can modify the symbols in the boxes;The machine worked using specific logical instructions formed by some and zeros that were put in the boxes. Turing showed that his machine, despite how simple it was, could perform any algorithmic problem that will be presented to him, with this theoretical concept, Turing was determined to make a machine that had a processing capacity like that of humans, he sought, he soughtthat this machine could perform different operations at the same time and at the same time, save information in a memory. Due to World War II, Turing had to put aside his desired machine and had to focus on howHe gave the victory to the allies and shortened the war from 2 to 4 years, saving millions of lives. After the war, Turing participated in the creation of the first computers: the “Colossus” project the first large -scale electronic calculator, then participated in the creation of “Manchester Mark I” and “Ace” the first computers with storage capacityby memory. After this, Turing devoted himself to the theoretical study and approach of artificial life or intelligence, and then committed to the accusations of his sexuality.
Hilbert, Gödel and the crisis of the foundations were only the bases that inspired the genius Alan Turing, thanks to him, we have everything we know today, the concepts he raised and its universal machine, however simple it is, is theBase of all technology, simple some and zeros, Turing machines, are what our computers and cell phones use day by day to be able to communicate from one point from the world to the other, they are what allows us to solve operations, they are the mostbasic that bases computing.
Right now it would be difficult to imagine a world without this level of technology, without a doubt the knowledge of these mathematicians and mainly those of Alan Turing were the ones that revealed all the potential of mathematics and in addition to this they allowed the world and technology to break the limitsThe impossible.
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