 In the 20th century , the study of mathematical logic provided the essential breakthrough that made artificial intelligence seem plausible . The foundations had been set by such works as Boole 's The Laws of Thought and Frege 's Begriffsschrift . Building on Frege 's system , Russell and Whitehead presented a formal treatment of the foundations of mathematics in their masterpiece , the Principia Mathematica in 1913 . Inspired by Russell 's success , David Hilbert challenged mathematicians of the 1920s and 30s to answer this fundamental question : " can all of mathematical reasoning be formalized ? " His question was answered by Gödel 's incompleteness proof , Turing 's machine and Church 's Lambda calculus . Their answer was surprising in two ways . 
