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Computing machines can’t be intelligent (...and Turing said so
 In Minds and Machines
, 2002
"... According to the conventional wisdom, Turing (1950) said that computing machines can be intelligent. I don’t believe it. I think that what Turing really said was that computing machines – computers limited to computing – can only fake intelligence. If we want computers to become genuinely intelligen ..."
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According to the conventional wisdom, Turing (1950) said that computing machines can be intelligent. I don’t believe it. I think that what Turing really said was that computing machines – computers limited to computing – can only fake intelligence. If we want computers to become genuinely intelligent, we will have to give them enough “initiative ” (Turing, 1948, p. 21) to do more than compute. In this paper, I want to try to develop this idea. I want to explain how giving computers more “initiative ” can allow them to do more than compute. And I want to say why I believe (and believe that Turing believed) that they will have to go beyond computation before they can become genuinely intelligent. 1. What I Think Turing Said People who try to make computers more intelligent say they are trying to produce “Artificial Intelligence ” (or “AI”). Presumably, they want the word “artificial ” to suggest that the intelligence they are trying to create will – like artificial vanilla – not have developed naturally. But some of their critics are convinced that anything that looks like intelligence in a computer will have to be artificial in another sense – the sense in which an artificial smile is artificial. Which is to say fake. Computers, they believe, cannot be genuinely intelligent because they lack a certain je ne sais quoi that genuine intelligence requires. The more extreme of these critics believe that what computers lack is fundamental. Perhaps they believe that intelligence requires an immortal soul. Perhaps they feel that it can only be implemented in flesh and blood. Perhaps they believe that it requires human experiences or human emotions. Such critics believe that computers cannot be genuinely intelligent, period. Other critics of AI are a bit more generous. They believe that computers cannot be genuinely intelligent until … Perhaps they believe that computers cannot be genuinely intelligent until they have access to better parallel processing or to special neural
Minimization and NP multifunctions
 Theoretical Computer Science
, 2000
"... . The implicit characterizations of the polynomialtime computable functions FP given by BellantoniCook and Leivant suggest that this class is the complexitytheoretic analog of the primitive recursive functions. Hence it is natural to add minimization operators to these characterizations and in ..."
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. The implicit characterizations of the polynomialtime computable functions FP given by BellantoniCook and Leivant suggest that this class is the complexitytheoretic analog of the primitive recursive functions. Hence it is natural to add minimization operators to these characterizations and investigate the resulting class of partial functions as a candidate for the analog of the partial recursive functions. We do so in this paper for Cobham's denition of FP by bounded recursion and for BellantoniCook's safe recursion and prove that the resulting classes capture exactly NPMV, the nondeterministic polynomialtime computable partial multifunctions. We also consider the relationship between our schemes and a notion of nondeterministic recursion dened by Leivant and show that the latter characterizes the total functions of NPMV. We view these results as giving evidence that NPMV is the appropriate analog of partial recursive. We reinforce this view by showing that for many ...
SelfReference and Validity revisited
"... This paper is a revision and expansion of my SelfReference and Validity, Synthese, 42, 1979, pp. 26574. The major changes consist in the addition of new //2, 4 and 7. The reader may be interested to see subsequent discussion of the issues in Priest and Routley 1982, Sorensen 1988, pp. 299310, Dra ..."
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This paper is a revision and expansion of my SelfReference and Validity, Synthese, 42, 1979, pp. 26574. The major changes consist in the addition of new //2, 4 and 7. The reader may be interested to see subsequent discussion of the issues in Priest and Routley 1982, Sorensen 1988, pp. 299310, Drange 1990 and Jacquette 19??
POSSIBLE mDIAGRAMS OF MODELS OF ARITHMETIC
"... Abstract. In this paper we investigate the complexity of mdiagrams of models of various completions of firstorder Peano Arithmetic (PA). We obtain characterizations that extend Solovay’s results for open diagrams of models of completions of PA. We first characterize the mdiagrams of models of Tru ..."
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Abstract. In this paper we investigate the complexity of mdiagrams of models of various completions of firstorder Peano Arithmetic (PA). We obtain characterizations that extend Solovay’s results for open diagrams of models of completions of PA. We first characterize the mdiagrams of models of True Arithmetic by showing that the degrees of mdiagrams of nonstandard models A of TA are the same for all m ≥ 0. Next, we obtain a more complicated characterization for arbitrary completions of PA. We then provide examples showing that some of the extra complication is needed. Lastly, we characterize sequences of Turing degrees that occur as (deg(T ∩ Σn))n∈ω, where T is a completion of PA. §1. Introduction. We use P (ω) to denote the class of all subsets of ω. Let LPA be the usual language of PA: relations +, ·, S, and <; and constants 0 and 1. We abbreviate True Arithmetic, the theory of the standard model of PA, by the initials TA. We use S n (0) to denote the numeral for n
INTERNAL AND EXTERNAL CONSISTENCY OF ARITHMETIC
"... What Gödel referred to as “outer ” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of ari ..."
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What Gödel referred to as “outer ” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation ” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of Hilbert’s finitist ideal which is achieved by going back to Kronecker’s programme of a general arithmetic of forms or homogeneous polynomials. The paper can be seen as a historical complement to our result on “The Internal Consistency of Arithmetic with Infinite Descent ” (Modern Logic, vol. 8, n ° 12, 2000, pp. 4786). An internal consistency proof for arithmetic means that transfinite induction is not needed and that arithmetic can be shown to be consistent within the bounds of arithmetic, that is with the help of Fermat’s infinite descent and Kronecker’s
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. ..."
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After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.
Derivation
, 2001
"... We return to the two firstorder formalizations of arithmetic to make some remarks on how the study of these subjects can be developed further. 1.1 Peano Arithmetic: Definability Since Firstorder Arithmetic is known to be rather intractable (it is undecidable), when looking for an effective approac ..."
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We return to the two firstorder formalizations of arithmetic to make some remarks on how the study of these subjects can be developed further. 1.1 Peano Arithmetic: Definability Since Firstorder Arithmetic is known to be rather intractable (it is undecidable), when looking for an effective approach to number theory, something which one might use for an automated theorem prover, it is natural to consider Peano Arithmetic since, as we said, all standard theorems of number theory which can be formulated as firstorder statements can be derived from PA. To see that this is true one needs to return to the formulations in the previous section and show that they can be used in PA. DEFINITION 1 (a) A relation r ⊆ ω n is defined in PA by the formula ϕ(x1,..., xn) if for each (k1,..., kn) ∈ ω n we have (k1,..., kn) ∈ r = ⇒ PA ⊢ ϕ ( ¯ k1,..., ¯ kn) (k1,..., kn) / ∈ r = ⇒ PA ⊢ ¬ϕ ( ¯ k1,..., ¯ kn) (b) A function f: ω n = ⇒ ω is defined in PA by the formula ϕ(x1,..., xn, y) if for each (k1,..., kn, k) ∈ ω n+1 we have f(k1,..., kn) = k = ⇒ PA ⊢ ϕ ( ¯ k1,..., ¯ kn, ¯ k) f(k1,..., kn) = k = ⇒ PA ⊢ ¬ϕ ( ¯ k1,..., ¯ kn, ¯ k) and PA ⊢ ∃!y ϕ ( ¯ k1,..., ¯ kn, y) In his 1931 paper Gödel showed that the relations and functions discussed in the previous section on firstorder arithmetic are indeed definable in PA 1. Using this expressive power of PA Gödel went on to give explicit firstorder 1 As are all decidable relations and computable functions. 1 sentences which are true in ω but cannot be derived from PA.
ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS
, 2010
"... Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics an ..."
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Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the BolzanoWeierstrass, HahnBanach, and intermediate value theorems, and then the implications of these arguments for such “crown jewels ” of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser [51] in his extension of Gödel’s incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him. Acknowledgements: I thank K. Vela Velupillai most particularly for his efforts to push me to consider these matters in the most serious manner, as well as my late father, J. Barkley Rosser [Sr.] and also his friend, the late Stephen C. Kleene, for their personal remarks on these matters to me over a long period of time. I also wish to thank Eric Bach, Ken Binmore, Herb Gintis, Jerome Keisler, Roger Koppl, David Levy, and Adrian Mathias for useful comments. The usual caveat holds. I also wish to dedicate this to K. Vela Velupillai who inspired it with his insistence that I finally deal with the work and thought of my father, J. Barkley Rosser [Sr.], as well as ShuHeng Chen, who supported him in this insistence. I thank both of them for this.