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Gödel on computability
"... Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of t ..."
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Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of the second theorem was explicitly based on Turing’s 1936 reduction of finite procedures to machine computations. Turing gave in his 1954 an understated analysis of finite procedures in terms of Post production systems. This analysis, prima facie quite different from that given in 1936, served as the basis for an exposition of various unsolvable problems. Turing had addressed issues of mentality and intelligence in contemporaneous essays, the best known of which is of course Computing machinery and intelligence. Gödel’s and Turing’s considerations from this period intersect through their attempt, on the one hand, to analyze finite, mechanical procedures and, on the other hand, to approach mental phenomena in a scientific way. Neuroscience or brain science was an important component of the latter for both: Gödel’s remarks in the Gibbs Lecture as well as in his later conversations with Wang and Turing’s Intelligent Machinery can serve as clear evidence for that. 2 Both men were convinced that some mental processes are not mechanical, in the sense that Turing machines cannot mimic them. For Gödel, such processes were to be found in mathematical experience and he was led to the conclusion that mind is separate from matter. Turing simply noted that for a machine or a brain it is not enough to be converted into a universal (Turing) machine in order to become intelligent: “discipline”, the characteristic
Lecture Notes on Natural Deduction 15816: Linear Logic
, 2012
"... We now turn our attention from the sequent calculus to natural deduction. As we saw, sequent calculus proofs lend themselves to an interpretation as sessiontyped concurrent processes. Generally speaking, natural deduction is related to the λcalculus and therefore to functional computation. In the ..."
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We now turn our attention from the sequent calculus to natural deduction. As we saw, sequent calculus proofs lend themselves to an interpretation as sessiontyped concurrent processes. Generally speaking, natural deduction is related to the λcalculus and therefore to functional computation. In the next lecture we will bridge the gap between the two. 1 Linear Hypothetical Judgments A linear hypothetical judgment has the form x1:A1,..., xn:An ⊢ A where we continue to use ∆ to denote the hypotheses. All the hypotheses must be labeled uniquely. Their order does not matter, but in accordance with linear logic each must be used in a proof exactly once. Unlike the sequent calculus, there are no left rules. Instead, the only access to the linear hypotheses is via a hypothesis rule x:A ⊢ A hyp The counterpart to the hypothesis rule is substitution. Substitution. If ∆ ⊢ A and ∆ ′ , x:A ⊢ C then ∆, ∆ ′ ⊢ C. This is called substitution because it describes an operation on proofs where the proof of ∆ ⊢ A is substituted for uses of the hypothesis x:A in the proof LECTURE NOTES FEBRUARY 20, 2012Natural Deduction L10.2 of ∆ ′ , x:A ⊢ C. We will see this more clearly when we introduce terms representing proofs. Unlike cut in the sequent calculus, substitution is rarely seen as a rule. We can also write substitution as an admissible rule: