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Lucas: Minds, Machines and Goedel

Lucas: Minds, Machines and Goedel. Are the limits on computation limits on the computational theory of mind?. Precursors to Goedel’s Proof. Epimenides Paradox: “All Cretans are Liars” “This statement is false” “This statement has five words”

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Lucas: Minds, Machines and Goedel

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  1. Lucas: Minds, Machines and Goedel Are the limits on computation limits on the computational theory of mind?

  2. Precursors to Goedel’s Proof • Epimenides Paradox: “All Cretans are Liars” • “This statement is false” • “This statement has five words” • The set of all sets that do not contain themselves as subsets • Halting problems

  3. Goedel’s Theorem • True of formal systems strong enough to include arithmetic • Completeness, Consistency • Goedel-numbering • “There is no number such that it is the Goedel number of the proof of the statement with Goedel number X” [X] • (“I am not provable.”)

  4. Lucas’s Application of Goedel’s Proof • Suppose that all mental processes are computations • We know you cannot compute the truth of a Goedel statement • Yet we know the statement is true • That knowledge is not computable • Hence not all mental processes are computations

  5. Penrose’s Extension • Creative mathematicians “see” the truth of theorems before they can prove them • Classical vs. “Quantum Computation” • Russell/James on “mathematical intuition” • “The Smell of Petroleum Pervades Throughout”

  6. Seeing, Knowing, Proving, Computing • Seeing the truth of “I am not computable” • Knowing the truth of “I am not computable” • Proving the truth of “I am not computable” • Computing the truth of “I am not computable” • Computability vs. Computationality • Feeling vs. Proving

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