Computability and human symbolic output
Languages of publication
This paper concerns “human symbolic output,” or strings of characters produced by humans in our various symbolic systems; e.g., sentences in a natural language, mathematical propositions, and so on. One can form a set that consists of all of the strings of characters that have been produced by at least one human up to any given moment in human history. We argue that at any particular moment in human history, even at moments in the distant future, this set is finite. But then, given fundamental results in recursion theory, the set will also be recursive, recursively enumerable, axiomatizable, and could be the output of a Turing machine. We then argue that it is impossible to produce a string of symbols that humans could possibly produce but no Turing machine could. Moreover, we show that any given string of symbols that we could produce could also be the output of a Turing machine. Our arguments have implications for Hilbert’s sixth problem and the possibility of axiomatizing particular sciences, they undermine at least two distinct arguments against the possibility of Artificial Intelligence, and they entail that expert systems that are the equals of human experts are possible, and so at least one of the goals of Artificial Intelligence can be realized, at least in principle.
- Corry, L., “On the origins of Hilbert’s sixth problem: physics and the empiricist approach to axiomatization”, pages 1697–1718 in Proceedings of the International Congress of Mathematicians, Madrid, Spain 2006. DOI: 10.4171/022-3/82
- Craig, W., “On axiomatizability within a system”, The Journal of Symbolic Logic, 18 (1953): 30–32. DOI: 10.2307/2266324
- Fodor, J., The Language of Thought, New York: Thomas Crowell 1975.
- Gödel, K, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”,Monash. Math. Phys., 38 (1931): 173–198.
- Gnedenko, J., “Zum sechsten Hilbertschen Problem”, pages 144–147 in Die Hilbertsche Probleme (ed. by P. Alexandrov), Ostwalds Klassiker der exakten Wissenschaften, Leipzig 1979.
- Hilbert, D., “Mathematische Probleme”, pages 253–297 in Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Klasse, 1900.
- Lucas, J.R., “Minds, machines and Gödel”, Philosophy, 36 (1961): 112–127.
- Lucas, J.R, “Mind, machines and Gödel: A retrospect”, a paper read to the Turing Conference at Brighton on April 6 th , 1990. http://users.ox.ac.uk/~jrlucas/Godel/brighton.html
- Lucas, J.R., “The Godelian argument: Turn over the page”, BSPS conference, Oxford, 1996. http://users.ox.ac.uk/~jrlucas/Godel/turn.html
- Megill, J., T. Melvin, and A. Beal, “On some properties of humanly known and humanly knowable mathematics”, Axiomathes (forthcoming).
- Penrose, R., The Emperor’s New Mind, Oxford: Oxford University Press 1989.
- Penrose, R., Shadows of the Mind, Oxford: Oxford University Press 1994.
- Searle, J., “Minds, brains and programs”, Behavioral and Brain Sciences, 3 (1980): 417–457. DOI: 10.1017/S0140525X00005756
- Wang, H., Popular Lectures on Mathematical Logic, Mineolam NY: Dover 1981.
- Wightman, A.S., “Hilbert’s sixth problem: Mathematical treatment of the axioms of physics”, pages 147–240 in Mathematical Developments Arising from Hilbert Problems (ed. by F.E. Browder), Symposia in Pure Mathematics, Amer. Math. Soc., Providence, RI 1976.
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