Argument Lucasa-Penrose’a

Argument Lucasa-Penrose’a – argument wykorzystujący twierdzenie Gödla o niezupełności systemów formalnych w celu pokazania, że umysłu nie można wyjaśnić w kategoriach czysto mechanistycznych. Argument został przedstawiony w różnych formach przez samego Kurta Gödla^[1], filozofów: Johna R. Lucasa^[2], Ernesta Nagela i Jamesa R. Newmana^[3] oraz fizyka Rogera Penrose’a^[4].

W każdym przypadku idea argumentu opiera się na drugim twierdzeniu Gödla. Jeśli można uznać zbiór S twierdzeń arytmetycznych dowodzonych przez system formalny F za zbiór twierdzeń prawdziwych, to na tej samej podstawie należy uznać prawdziwość twierdzenia arytmetycznego równoważnego spójności tego systemu Ω(F), bowiem gdyby system F nie był spójny, to zbiór S obejmowałby dowolne zdanie wyrażalne w F, w tym zdania wzajemnie sprzeczne. Jednak na mocy drugiego twierdzenia Gödla zdanie Ω(F) nie jest wyprowadzalne w F, jeśli jest prawdziwe, zatem nie jest możliwe pokrycie wszystkich funkcji ludzkiego umysłu (w szczególności kryteriów prawdziwości twierdzeń arytmetycznych) za pomocą jakiegokolwiek systemu formalnego F.

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Przypisy[edytuj | edytuj kod]

↑ Wang 1997 ↓, „My incompleteness theorem makes it likely that mind is not mechanical, or else mind cannot understand its own mechanism.”, s. 186.
↑ John R. Lucas. Mind, machines and Gödel w A.R. Anderson (red.) Minds and Machines, s. 120–124. Englewood Cliffs, 1964. „Gödel’s theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines.”.
↑ Nagel i Newman 1958 ↓, „Gödel's conclusions bear on the question whether a calculating machine can be constructed that would match the human brain in mathematical intelligence. Today’s calculating machines have a fixed set of directives built into them; these directives correspond to the fixed rules of inference of formalized axiomatic procedure. The machines thus supply answers to problems by operating in a step-by-step manner, each step being controlled by the built-in directives. But, as Gödel showed in his incompleteness theorem, there are innumerable problems in elementary number theory that fall outside the scope of a fixed axiomatic method, and that such engines are incapable of answering, however intricate and ingenious their built-in mechanisms may be and however rapid their operations. Given a definite problem, a machine of this type might be built for solving it; but no one such machine can be built for solving every problem. The human brain may, to be sure, have built-in limitations of its own, and there may be mathematical problems it is incapable of solving. But, even so, the brain appears to embody a structure of rules of operation which is far more powerful than the structure of currently conceived artificial machines. There is no immediate prospect of replacing the human mind by robots.”, s. 78.
↑ Roger Penrose. Mathematical intelligence w Jean Khalfa (red.) What is Intelligence?, s. 107–136. Cambridge University Press, Cambridge, United Kingdom, 1994. „The inescapable conclusion seems to be: Mathematicians are not using a knowably sound calculation procedure in order to ascertain mathematical truth. We deduce that mathematical understanding – the means whereby mathematicians arrive at their conclusions with respect to mathematical truth – cannot be reduced to blind calculation.”.

Bibliografia[edytuj | edytuj kod]

Hao Wang: A Logical Journey: From Gödel to Philosophy. A Bradford Book, 1997. ISBN 978-0-262-23189-3.
Ernest Nagel, James R. Newman: Gödel’s Proof. London: Routledge and Kegan Paul, 1958. ISBN 0-415-35528-1.

Linki zewnętrzne[edytuj | edytuj kod]

JasonJ. Megill JasonJ., The Lucas-Penrose Argument about Gödel’s Theorem, Internet Encyclopedia of Philosophy, ISSN 2161-0002 [dostęp 2018-06-27] (ang.).

[CITEREFWang1997„My_incompleteness_theorem_makes_it_likely_that_mind_is_not_mechanical,_or_else_mind_cannot_understand_its_own_mechanism.”186-1] Wang 1997 ↓, „My incompleteness theorem makes it likely that mind is not mechanical, or else mind cannot understand its own mechanism.”, s. 186.

[2] John R. Lucas. Mind, machines and Gödel w A.R. Anderson (red.) Minds and Machines, s. 120–124. Englewood Cliffs, 1964. „Gödel’s theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines.”.

[CITEREFNagelNewman1958„Gödel's_conclusions_bear_on_the_question_whether_a_calculating_machine_can_be_constructed_that_would_match_the_human_brain_in_mathematical_intelligence._Today’s_calculating_machines_have_a_fixed_set_of_directives_built_into_them;_these_directives_correspond_to_the_fixed_rules_of_inference_of_formalized_axiomatic_procedure._The_machines_thus_supply_answers_to_problems_by_operating_in_a_step-by-step_manner,_each_step_being_controlled_by_the_built-in_directives._But,_as_Gödel_showed_in_his_incompleteness_theorem,_there_are_innumerable_problems_in_elementary_number_theory_that_fall_outside_the_scope_of_a_fixed_axiomatic_method,_and_that_such_engines_are_incapable_of_answering,_however_intricate_and_ingenious_their_built-in_mechanisms_may_be_and_however_rapid_their_operations._Given_a_definite_problem,_a_machine_of_this_type_might_be_built_for_solving_it;_but_no_one_such_machine_can_be_built_for_solving_every_problem._The_human_brain_may,_to_be_sure,_have_built-in_limitations_of_it] Nagel i Newman 1958 ↓, „Gödel's conclusions bear on the question whether a calculating machine can be constructed that would match the human brain in mathematical intelligence. Today’s calculating machines have a fixed set of directives built into them; these directives correspond to the fixed rules of inference of formalized axiomatic procedure. The machines thus supply answers to problems by operating in a step-by-step manner, each step being controlled by the built-in directives. But, as Gödel showed in his incompleteness theorem, there are innumerable problems in elementary number theory that fall outside the scope of a fixed axiomatic method, and that such engines are incapable of answering, however intricate and ingenious their built-in mechanisms may be and however rapid their operations. Given a definite problem, a machine of this type might be built for solving it; but no one such machine can be built for solving every problem. The human brain may, to be sure, have built-in limitations of its own, and there may be mathematical problems it is incapable of solving. But, even so, the brain appears to embody a structure of rules of operation which is far more powerful than the structure of currently conceived artificial machines. There is no immediate prospect of replacing the human mind by robots.”, s. 78.

[4] Roger Penrose. Mathematical intelligence w Jean Khalfa (red.) What is Intelligence?, s. 107–136. Cambridge University Press, Cambridge, United Kingdom, 1994. „The inescapable conclusion seems to be: Mathematicians are not using a knowably sound calculation procedure in order to ascertain mathematical truth. We deduce that mathematical understanding – the means whereby mathematicians arrive at their conclusions with respect to mathematical truth – cannot be reduced to blind calculation.”.

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