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Thinking May Be More Than Computing
 Cognition
, 1986
"... The uncomputable parts of thinking (if there are any) can be studied in much the same spirit that Turing (1950) suggested for the study of its computable parts. We can develop precise accounts of cognitive processes that, although they involve more than computing, can still be modelled on the machin ..."
Abstract

Cited by 18 (3 self)
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The uncomputable parts of thinking (if there are any) can be studied in much the same spirit that Turing (1950) suggested for the study of its computable parts. We can develop precise accounts of cognitive processes that, although they involve more than computing, can still be modelled on the machines we call ‘computers’. In this paper, I want to suggest some ways that this might be done, using ideas from the mathematical theory of uncomputability (or Recursion Theory). And I want to suggest some uses to which the resulting models might be put. (The reader more interested in the models and their uses than the mathematics and its theorems, might want to skim or skip the mathematical parts.) 1.
Induction, Pure and Simple
 INFORMATION AND CONTROL 35, 276336 (1977)
, 1977
"... Induction is the process by which we reason from the particular to the general; In this paper we use ideas from the theory of abstract machines and recursion theory to study this process. We focus on pure induction in which the conclusions "go beyond the information given " in the premises ..."
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Cited by 13 (7 self)
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Induction is the process by which we reason from the particular to the general; In this paper we use ideas from the theory of abstract machines and recursion theory to study this process. We focus on pure induction in which the conclusions "go beyond the information given " in the premises from which they are derived and on simple induction, which is rather a stark kind of induction that deals with computable predicates on the integers in rather straightforward ways. Our basic question is "What are the relationships between the kinds of abstract machinery we bring to bear on the job of doing induction and our ability to do that job well? " Our conclusions are as follows: (1) If we use only the abstract machinery of the digital computer in a computing center (which we assume to be capable of only evaluating totally computable functionals or functionals in 210 of the Arithmetic Hierarchy) then a single inductive procedure can only develop finitely many sound theories. (2) If we use only the abstract machinery of the mathematician (which we assume to be the machinery required to evaluate a functional in 271 of the Arithmetic Hierarchy) then we can develop inductive
Kleene’s Amazing Second Recursion Theorem (Extended Abstract)
"... This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number ..."
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Cited by 1 (1 self)
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This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene (1938). In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows: Theorem 1 (SRT). Fix a set V ⊆ N, and suppose that for each natural number n ∈ N = {0, 1, 2,...}, ϕ n: N n+1 ⇀ V is a recursive partial function of (n + 1) arguments with values in V so that the standard assumptions (1) and (2) hold with {e}(⃗x) = ϕ n e (⃗x) = ϕ n (e, ⃗x) (⃗x = (x1,..., xn) ∈ N n). (1) Every nary recursive partial function with values in V is ϕ n e for some e. (2) For all m, n, there is a recursive (total) function S = S m n: N m+1 → N such that {S(e, ⃗y)}(⃗x) = {e}(⃗y, ⃗x) (e ∈ N, ⃗y ∈ N m, ⃗x ∈ N n). Then, for every recursive, partial function f(e, ⃗y, ⃗x) of (1+m+n) arguments with values in V, there is a total recursive function ˜z(⃗y) of m arguments such that
1 Introduction Degrees of Unsolvability
, 2006
"... Modern computability theory began with Turing [Turing, 1936], where he introduced ..."
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Modern computability theory began with Turing [Turing, 1936], where he introduced