 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 . 
 In the 20th century , the study of mathematical logic provided the essential breakthrough <a href='null_relcl' title='1'><mark>that</mark></a>made artificial intelligence seem plausible . The foundations <a href='been_done' title='2'><mark>had</mark></a> been set by such works as Boole <a href='null_genetive' title='3'><mark>'s</mark></a>The Laws of Thought and Frege <a href='null_genetive' title='4'><mark>'s</mark></a>Begriffsschrift . Building on Frege <a href='null_genetive' title='5'><mark>'s</mark></a>system , Russell and Whitehead presented a formal treatment of the foundations of mathematics in their masterpiece , the Principia Mathematica in 1913 . Inspired by Russell <a href='null_genetive' title='6'><mark>'s</mark></a>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 <a href='null_genetive' title='7'><mark>'s</mark></a>incompleteness proof , Turing <a href='null_genetive' title='8'><mark>'s</mark></a>machine and Church <a href='null_genetive' title='9'><mark>'s</mark></a>Lambda calculus . Their answer was surprising in two ways . 
In the 20th century , the study of mathematical logic provided da essential breakthrough made artificial intelligence seem plausible . Tdafoundations done been set by such works as Boole  The Laws of Thought and Frege  Begriffsschrift . Building on Frege  system , Russell and Whitehead presented a formal treatment of the foundations of mathematics in their masterpiece , the Principia Mathematica in 1913 . Inspired by Russell  success , David Hilbert challenged mathematicians of the 1920s and 30s to answer dis fundamental question : " can all of mathematical reasoning b formalized ? " His question was answered by Gödel  incompleteness proof , Turing  machine and Church  Lambda calculus . Their answer was surprising in 2 ways .