【4月21日】名家讲座第33讲：“On the Question of Whether the Mind Can be Mechanized”——逻辑与哲学系列讲座
In this talk I will discuss the question of whether Gödel's incompleteness theorems imply that “the mind cannot be mechanized”. The story begins with Gödel. He argued for a weaker, disjunctive conclusion to the effect that the incompleteness theorems imply that either “the mind cannot be mechanized” or that “there are absolutely undecidable statements.” Since then others -- most notably Lucas and Penrose -- have argued for the stronger conclusion, namely, that the incompleteness theorems imply the first of these two disjuncts -- that “the mind cannot be mechanized”.
In the first part of the talk I will sharpen the underlying notions. This will enable us to pull the discussion into a setting where definitive results can be proved. We shall see that (thus formalized) Gödel's disjunction is indeed provable. This then leads to the question: “Which disjunct holds?” I will discuss some recent independence results which show that when (thus formalized) one can show that although the disjunction is provable neither disjunct is provable or refutable. Moreover, the result is robust in that it persists when one strengthens the underlying principles of knowledge. I will conclude that the questions of whether “the mind can be mechanized” or whether “there are absolutely undecidable statements” are themselves good candidates for statements that are “absolutely undecidable”.