David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Minds and Machines 13 (1):23-48 (2003)
This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical
objection” to his view that machines can think. Logico-mathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it
should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human
mathematicians presumably do.
|Keywords||Artificial Intelligence Church-Turing Thesis Computability Incompleteness Ordinal Logics Undecidability Turing effective procedure mathematical objection Undecidability|
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Gualtiero Piccinini (2007). Computing Mechanisms. Philosophy of Science 74 (4):501-526.
Gualtiero Piccinini (2008). Computers. Pacific Philosophical Quarterly 89 (1):32–73.
Gualtiero Piccinini (2007). Computational Modeling Vs. Computational Explanation: Is Everything a Turing Machine, and Does It Matter to the Philosophy of Mind? Australasian Journal of Philosophy 85 (1):93 – 115.
Gualtiero Piccinini (2010). The Mind as Neural Software? Understanding Functionalism, Computationalism, and Computational Functionalism. Philosophy and Phenomenological Research 81 (2):269-311.
Gualtiero Piccinini (2004). Functionalism, Computationalism, & Mental States. Studies in the History and Philosophy of Science 35 (4):811-833.
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