One of the topics in mathematics that I always find fascinating is logics. I can find myself reading and practicing few problems in logics, sometimes getting nice and interesting results, sometimes ending with a headache.
Some days ago, reading the (by the way, great!) book Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem, I found an aspect in the history of logics that takes me time to reflect every time I read about it: the Russell's Paradox.
Some days ago, reading the (by the way, great!) book Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem, I found an aspect in the history of logics that takes me time to reflect every time I read about it: the Russell's Paradox.
Everything started with an incoherent thought. Bertrand Russell (British philosopher, logician, mathematician, historian, writer, social critic, political activist, Nobel laureate in Litereature-1950) was the person who got that incoherent thought. It is known as the Russell's paradox and can be explained with the tale of the librarian:
One day, a librarian discovers a collection of catalogs. There are different catalogs, for novels, essays, tales, etc. The librarian finds that some catalogs include themselves and some others not. In order to simplify the system, the librarian elaborates two new catalogs, one catalog who includes catalogs that includes themselves, and another catalog who doesn't. After completing this task, the librarian finds out a problem: the catalog that includes the list of the catalogs that don't include themselves, should it catalog itself? If it does, by definition it shouldn't be included, whereas if it doesn't, it should includes itself by definition. The librarian is in impossible situation!
This incoherency was a consequence of working with mathematical axioms, which until that time they were considered as evident and good enough to define all the mathematics. A way to tackle the problem was to create an additional axiom that prohibits to any group be a member of itself. This would reduce the Russell's paradox and converts it as superfluous the question about including in the catalog those who are not included themselves.
Russell expend the next decade examining the axioms. In 1910, together with Alfred Whitehead (English mathematician and philosopher), published the first of three volumes of Principia Mathematica, trying the problem that came up from his own paradox. Apparently, the building of a coherent logics, strong enough to answer any question, was on a good track.
One day, a librarian discovers a collection of catalogs. There are different catalogs, for novels, essays, tales, etc. The librarian finds that some catalogs include themselves and some others not. In order to simplify the system, the librarian elaborates two new catalogs, one catalog who includes catalogs that includes themselves, and another catalog who doesn't. After completing this task, the librarian finds out a problem: the catalog that includes the list of the catalogs that don't include themselves, should it catalog itself? If it does, by definition it shouldn't be included, whereas if it doesn't, it should includes itself by definition. The librarian is in impossible situation!
This incoherency was a consequence of working with mathematical axioms, which until that time they were considered as evident and good enough to define all the mathematics. A way to tackle the problem was to create an additional axiom that prohibits to any group be a member of itself. This would reduce the Russell's paradox and converts it as superfluous the question about including in the catalog those who are not included themselves.
Russell expend the next decade examining the axioms. In 1910, together with Alfred Whitehead (English mathematician and philosopher), published the first of three volumes of Principia Mathematica, trying the problem that came up from his own paradox. Apparently, the building of a coherent logics, strong enough to answer any question, was on a good track.
Then, an unknow 25 years old mathematician, came to say something new, something that would stop that thought of an ideal world created with a pure and perfect mathematical structure, this man was Kurt Gödel. But Gödel deserves a separate post.
See you around,
Jesús
See you around,
Jesús