COMPUTING AND MATHEMATICAL PARADOXES

COMPUTING AND MATHEMATICAL PARADOXES

1. Paradox

   • Something that at first glance seems to be false but that is actually true, or that it seems be true but is strictly false, or simply that it encloses in itself contradictions
     • Etymologically, paradox means what para ten doxan, "what goes in against public opinion ”.
       • a paradox is a proposition which is apparently true, but what leads to a logical contradiction or situation in which the sense is violated common.
         • Applications
           • It greatly stimulates reflection, by point that philosophers use it frequently to reveal complexity of a reality
           • I know used to make the limitations palpable possessing the tools of the mind

2. Logical paradox

   • Bertrand Russell
     • one of the key figures in the question of mathematical paradoxes
     • He found cases in which trials, apparently impeccable, they led to contradictions; its contribution was essential to spread the idea that these contradictions were the causing serious crises and that should be resolved in some way.
     • Russell's paradox has its origin in another of ancient Greece, although not applied to the set theory, known as Paradox of Epimenides or of the liar: This claim is false! If the assertion is false, then it must be true; but yes it is true then it is false

3. Gilbert's rescue idea

   • Hilbert had the idea of ​​creating a language artificial perfect for reasoning, deduction and mathematics, and emphasized in which the axiomatic method was very important, since with him it was possible to leave of a set of ―postulated axioms basic - and well-formed formulas, to deduce and derive effective theorems.
     • Would create a discipline totally new: metamathematics, a reflective area of mathematics dedicated to studying what this may or may not reach
       • Another issue why your idea was not possible, is that if a fully formal axiomatic system, its meaning loses validity, and its consideration is limited to a simple game in which your pieces will be marks on a paper of the that it is possible to deduce theorems from axioms
         • to study the mathematics through methods mathematicians, what is required is to extract its meaning, and limited to the study of a structured artificial language through absolutely precise rules.
           • The proposal of Hilbert sought to create rules so precise that the demonstrations would only have to submit to an impartial verdict, in the that a mechanical procedure confirmed the veracity or not of the claim, and not would have appeal.

4. GÖDEL AND INCOMPLETITUDE

   • Gödel ingeniously resorted to paradox of the liar for the demonstration of the incompleteness, and constructed a statement that says of itself that it is unprovable.
     • Your demo uses the recursion in many of the functions that they work on lists, precisely the same what LISP does.
       • In 1931 there were no computers or programming languages, in hindsight this article reflects clearly a language with the structure of the programming that many years later began to use.

5. THE TURING MACHINE

   • In 1936
     • Alan Turing discovered what would later be known as non-computability.
       • Like Gödel's article, Turing's describes what is now known as language programming
         • It is rather a Rather rudimentary machine language.
           • His idea was a machine that was capable of run, on a tape infinitely long, box-by-box operations. Saying device could read the content of each tape box and, in accordance with the internal state of the machine, would modify it Or I'd leave it the same, then run the tape a space to one side in a process repetitive.