How Godel’s theorem shaking stability of mathematical reasoning
The barber of Sevilla is “one who shaves all those, and those only, who do not shave themselves.” Then, the confusing question is if he shaves himself.
In case he does, he cannot be a barber because a barber does not shave anyone who shaves himself, according to the above-mentioned sentence.
Even if he doesn’t, he still cannot be a barber because he falls in the category of those who do not shave themselves.
The so-called barber paradox, which was used by British mathematician and philosopher Bertrand Russell, comes up with a problem with self-reference in logic.
There are many similar examples like the Epimenides paradox.
Epimenides once said that “all Cretans are liars.” As Epimenides himself was Cretan, however, the sentence does not make sense whether he is lying or not.
A question of “can omnipotent God create an immovable stone” or an axiom like “there is no rule without exception” are the same self-referential paradoxes.
Why are they significant? Since they show the basic idea of the incompleteness theorem, which Austrian logician Kurt Godel published in 1931 to greatly affect both mathematics and philosophy.
The theorem shows the defects inherent in formal systems, which depend on rules and logical calculus _ it is impossible to find a complete and consistent set of axioms for all mathematics.
Rebecca Goldstein’s book, dubbed “Incompleteness: The Proof and Paradox of Kurt Godel,” probes the life and work of Godel while showing how his work shook the stability of mathematical reasoning.
Because computers rely on formal systems, the author also points out that they will have limitations posed by the incompleteness theorem.
In other words, the theorem entails “the falsity of mechanism, the dead-endedness of the field of artificial intelligence.”
She might be right when the best-selling book was published in 2015.
But it remains to be seen whether Goldstein’s analysis still holds after AI defeats its human competitors in a mounting number of areas.