Why Gödel's theorem cannot refute computationalism

LaForte, Geoffrey; Hayes, Patrick J.; Ford, Kenneth M.

doi:10.1016/s0004-3702(98)00052-6

Cited by 20 publications

(13 citation statements)

References 6 publications

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“…But their arguments assume undefined capabilities of human reasoning and permit humans to participate within those arguments, which are key reasons the form of reasoning in their arguments is fallacious [LaForte et al 1998]. In contrast, part (4) of Definition 8.6 captures a modest but sufficient deductive requirement within a rigorous framework.…”

Section: Relation To Previous Resultsmentioning

confidence: 99%

“…There is no known definition of such an agent in ZF, so such arguments are inherently nonrigorous according to the standard criterion for rigor. Indeed, a recent investigation concludes that the form of those arguments used by Lucas and Penrose can also be used to obtain paradoxical • A. Charlesworth conclusions [LaForte et al 1998]. Gödel's Incompleteness Theorem and Tarski's Theorem on the undefinability of truth demonstrate how a proper mathematical model permits us to obtain rigorous results, even using arguments inspired by the classic paradoxes.…”

Section: Ifmentioning

confidence: 99%

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Comprehending software correctness implies comprehending an intelligence-related limitation

Charlesworth

2006

ACM Trans. Comput. Logic

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This article applies mathematical logic to obtain a rigorous foundation for previous inherently nonrigorous results and also extends those previous results. Roughly speaking, our main theorem states: any agent A that comprehends the correctness-related properties of software S also comprehends an intelligence-related limitation of S. The theorem treats the output of S, if any, as an attempt at solving a halting problem. Previous nonrigorous attempts to obtain similar theorems depend on infallibility assumptions on both the agent and the software. The hypothesis that intelligent agents and intelligent software must be infallible has been widely questioned. In addition, recent work by others has determined that well-known previous attempts use a fallacious form of reasoning; that is, the same form of reasoning can yield paradoxical results. Our main theorem avoids infallibility assumptions on both the agent and the software. In addition, our proof is rigorous, in the sense that in principle one can carry it out in Zermelo-Fraenkel set theory. The software correctness framework considered in the main theorem is that of Hoare logic.

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Section: Relation To Previous Resultsmentioning

confidence: 99%

Section: Ifmentioning

confidence: 99%

Comprehending software correctness implies comprehending an intelligence-related limitation

Charlesworth

2006

ACM Trans. Comput. Logic

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“…Penrose (1994Penrose ( , 1997 argued on the basis of Gödel's first incompleteness theorem, which shows the incompleteness of any consistent formal system for arithmetic that mathematical insight is fundamentally noncomputable, and therefore requires the OR phenomenon and associated quantum computational processing in the brain. Numerous respondents have demonstrated, however, that Gödel's theorem does not have the implications drawn by Penrose (e.g., Grush & P. S. Churchland, 1995;LaForte, Hayes, & Ford, 1998;Manaster-Ramer, Zadrozny, & Savitch, 1990;Shapiro, 2003). Although we are still far from having a neurocomputational theory of mathematical reasoning, Gödel's theorem does not imply that mathematical insight must be noncomputable.…”

Section: The Psychological Argumentmentioning

confidence: 99%

Is the Brain a Quantum Computer?

et al. 2006

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We argue that computation via quantum mechanical processes is irrelevant to explaining how brains produce thought, contrary to the ongoing speculations of many theorists. First, quantum effects do not have the temporal properties required for neural information processing. Second, there are substantial physical obstacles to any organic instantiation of quantum computation. Third, there is no psychological evidence that such mental phenomena as consciousness and mathematical thinking require explanation via quantum theory. We conclude that understanding brain function is unlikely to require quantum computation or similar mechanisms.

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“…For instance, this is made clear in an AI journal article (LaForte et al 1998), whose conclusions include the following, where we indicate in italics shortcomings of all arguments that have attempted to apply Gödel-Turing like results:…”

Section: Introductionmentioning

confidence: 96%

The Comprehensibility Theorem and the Foundations of Artificial Intelligence

Charlesworth

2014

Minds & Machines

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Problem-solving software that is not-necessarily infallible is central to AI. Such software whose correctness and incorrectness properties are deducible by agents is an issue at the foundations of AI. The Comprehensibility Theorem, which appeared in a journal for specialists in formal mathematical logic, might provide a limitation concerning this issue and might be applicable to any agents, regardless of whether the agents are artificial or natural. The present article, aimed at researchers interested in the foundations of AI, addresses many questions related to that theorem, including differences between it and results of Gödel and Turing that have sometimes played key roles in Minds and Machines articles. This study also suggests that-if one is willing to assume a thesis due to Donald Knuth-the Comprehensibility Theorem is the first mathematical theorem implying the impossibility of any AI agent or natural agent-including a not-necessarily infallible human agent-satisfying a rigorous and deductive interpretation of the self-comprehensibility challenge. Some have pointed out the difficulty of self-comprehensibility, even according to presumably a less rigorous interpretation. This includes Socrates, who considered it to be among the most important of intellectual tasks. Selfcomprehensibility in some form might be essential for a kind of self-reflection useful for self-improvement that might enable some agents to increase their success. We use the methods of applied mathematics, rather than philosophy, although some topics considered could be of interest to philosophers.

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Why Gödel's theorem cannot refute computationalism

Cited by 20 publications

References 6 publications

Comprehending software correctness implies comprehending an intelligence-related limitation

Comprehending software correctness implies comprehending an intelligence-related limitation

Is the Brain a Quantum Computer?

The Comprehensibility Theorem and the Foundations of Artificial Intelligence

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