The limits of human mathematics

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Volumn 35, Issue SUPPL. 15, 2001, Pages 93-117

  Salmon, Nathan ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

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EID: 84870818976     PISSN: 00294624     EISSN: 14680068     Source Type: Journal    
DOI: 10.1111/0029-4624.35.s15.5     Document Type: Article

References (21)

1. Nagel and Newman, Gödel's Proof (New York University Press, 1958, 1967), at pp. 100-102.
2. In The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford University Press, 1989), at p. 416.
3. Penrose revisits some of the issues in Shadows of the Mind: A Search for the Missing Science of Consciousness (Oxford University Press, 1994)
4. and "Beyond the Doubting of a Shadow: A Reply to Commentaries on Shadows of the Mind", Psyche, 2, 23 (1996).
5. Lucas, "Minds, Machines and Gödel", Philosophy, XXXVI (1961);
6. reprinted in A. R. Anderson, ed., Minds and Machines (Englewood Cliffs, NJ: Prentice-Hall, 1964), pp. 43-59, at 44, 47. A conclusion opposite in thrust from that of Lucas, Nagel and Newman
7. and Penrose is urged by Judson Webb, Mechanism, Mentalism and Metamathematics (Dordrecht: D. Reidel, 1980).
8. Putnam, "Minds and Machines", in Sidney Hook, ed., Dimensions of Mind: A Symposium (New York: New York University Press, 1960);
9. Lucas has replied, in "Minds, Machines, and Gödel: A Retrospect", in P. J. R. Millican and A. Clark, eds, Machines and Thought: The Legacy of Alan Turing, Volume 1(Oxford University Press, 1996), that the mechanist's claim that the proposed logistic system captures human mathematical reasoning is otiose unless the mechanist concedes that the system is consistent, and it is from this premise that Lucas derives the Gödelian sentence (p. 117). But unless the premise is itself proved mathematically, Lucas's derivation does not constitute a proof, or anything close to a proof. Given Lucas's objective, it is not sufficient for him to argue merely that mechanism cannot be proved.
10. An assessment of the arguments and assertions of Lucas and Penrose is provided in Stewart Shapiro, "Incompleteness, Mechanism, and Optimism", The Bulletin of Symbolic Logic, 4(September 1998), pp. 273-302.
11. "Some Basic Theorems on the Foundations of Mathematics and Their Implications", in Gödel's Collected Works, III: Unpublished Essays and Lectures, S. Feferman, J. W. Dawson, Jr., W. Goldfarb, C. Parsons, and R. N. Solovay, eds (Oxford University Press, 1995), pp. 304-323, at 310.
12. See also Hao Wang, A Logical Journey: From Gödel to Philosophy (Cambridge, Ma.: MIT Press, 1996), especially chapters 6 and 7, pp. 183-246.
13. Under this restriction, the deductive system that takes all sentences expressing truths of arithmetic as axioms (though it exists) is disqualified as a logistic or formal system or theory. Thus Wang-the expositor who more than any other brought Gödel's philosophical views into the public domain-gives the following informal statement of the first incompleteness theorem (op. cit., p. 3): No formal system of mathematics can be both consistent and complete; or alternatively, Any consistent formal theory of mathematics must contain undecidable propositions.
14. Similarly, C. Smorynski, "The Incompleteness Theorems", in J. Barwise, ed., Handbook of Mathematical Logic (Amsterdam: North Holland, 1977, 1983), pp. 821-865, states the theorem as follows: Let T be a formal theory containing arithmetic. Then there is a sentence φ which asserts its own unprovability and is [undecidable by T if T is ωo-consistent] (p. 825).
15. cons roughly for any logistic system suitable for arithmetic that includes the resources to designate any recursive function of integers and whose primitive deductive basis is recursive. For details, see Elliot Mendelson, Introduction to Mathematical Logic (New York: D. Van Nostrand, 1979), chapter 3, especially pp. 161-162.
16. Rosser, "Extensions of Some Theorems of Gödel and Church", Journal of Symbolic Logic, 1(1936), pp. 87-91.
17. It follows from the result obtained by William Craig in "Axiomatizability Within a System", Journal of Symbolic Logic, 18, 1(March 1953), pp. 30-32, that if Ax is recursively enumerable, then even if Ax is not itself recursive, Hu Math is primitive recursively axiomatizable. (Thanks to C. Anthony Anderson for calling my attention to the relevance of Craig's result.)
18. Church, Introduction to Mathematical Logic, I (Princeton University Press, 1956), at pp. 50-51.
19. C. Anthony Anderson makes a related objection in "Alonzo Church's Contributions to Philosophy and Intensional Logic", Bulletin of Symbolic Logic, 4, 2(June 1998), pp. 129-171, at 130-131.
20. Cf. Lewis Carroll, "What the Tortoise Said to Achilles", Mind, N. S. IV, 14(April 1895), pp. 278-280.
21. Wang, From Mathematics to Philosophy (London: Routledge and Kegan Paul, 1974), at pp. 324-326.