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By Haskell B. Curry, Robert Feys, William Craig, A. Heyting, A. Robinson

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Ca. 1951, 475) • The importance of instruction modification: “What we want is a machine that can learn from experience. The possibility of letting the machine alter its own instructions provides the mechanism for this. ” (1947, 393) Instruction-modification leads from one Turing machine to another; and underpins the central feature of the Multi-Machine theory of mind, the transformation of one Turing machine into another. Different instruction table, different Turing machine. In a lecture given circa 1951, Turing made it clear that his—then radical— idea that machines can learn is the crux of his reply to the Mathematical Objection; and he stressed the importance of the idea of learning new methods of proof in his 1947 discussion of the objection, describing the mathematician as searching around and finding new methods of proof.

Nevertheless, Georg Kreisel, reviewing our paper for Mathematical Reviews, did not mention this fact, but did write: ... 5 Yuri Matiyasevich’s Triumph In January 1970, I received a phone call from a friend informing me that JR had apparently been proved by a Russian mathematician. This was Yuri Matiyasevich, then twenty-two years old. He had provided a Diophantine definition for the set {< u, v > | v = F2 u } , where Fn is the nth Fibonacci number, defined as follows: F0 = 0, F1 = 1, Fn + 2 = Fn +1 + Fn .

Even Turing machines can compute uncomputable functions. In Unconventional Models of Computation, ed. C. Calude, J. Casti, and M. Dinneen, 150–164. London: Springer-Verlag. Copeland, B. J. 1998c. Super Turing-machines. Complexity 4:30–32. Copeland, B. J. 2000. Narrow versus wide mechanism. Journal of Philosophy 97:5–32. Reprinted in Computationalism: New Directions, ed. M. Scheutz, 59–86. Cambridge: MIT Press, 2002. Copeland, B. J. 2002a. Accelerating Turing machines. Minds and Machines 12:281–301.

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