Abstract
This paper presents a new perspective on the Lucas-Penrose arguments, which attempt to connect Gödel’s incompleteness theorems to the thesis of mechanism in the philosophy of mind. I begin by taking a close look at Hilary Putnam’s own response to Lucas-Penrose, which is widely taken to be decisive. I shall largely concur, but there is more to be learned. I will suggest that certain non-monotonic models of mathematical reasoning significantly alter the philosophical context for these arguments. I go on to describe a structural constraint on the rational coherence of the alternative cognitive evolutions allowed by these models. In the presence of that constraint, it is shown that if the evolutions allowed by such a model are mediated by an effective rule of revision, then the model is incapable of capturing certain inductive inferences of the most elementary kind.
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Notes
- 1.
J. Lucas, Minds, Machines and Gödel, Philosophy 36: 112–127.
- 2.
Putnam’s paper is reprinted in Mind, Language and Reality: Philosophical Papers, v. 2, Cambridge University Press, 1975. Boolos makes essentially the same claim about Putnam vis-à-vis Lucas–Penrose type arguments in his introductory note to Gödel’s Gibbs Lecture in Collected Works, v. III, p. 295. Note that Putnam’s ‘response’ was published before Lucas’s paper! Its actual target was the related discussion of the significance of the incompleteness theorems in the concluding sections of Nagel and Newman’s book Gödel’s Proof , reissued by New York University Press in 2001.
- 3.
I should mention, of course, that Penrose has not been silent on these matters in the intervening twenty-five years. For example, his later book Shadows of the Mind (Oxford, 1994), Penrose gives a variant of the argument concerning soundness rather than consistency. We shall touch on this variation below. Solomon Feferman has drawn attention to a number of technical problems in Penrose’s argument in Penrose’s Gödelian Argument, Psyche 2, 7 (1996), some of which play a role below.
- 4.
Shadows of the Mind, op, cit. What follows is a reconstruction, not a rendition, of Penrose’s later argument.
- 5.
Observe that since the sentence U above is a consequence of ConT in elementary arithmetics, we can also prove that U is true.
- 6.
For more or less similar arguments, see, for example, Stewart Shapiro, Truth and Proof – Through Thick and Thin, Journal of Philosophy 93, 493–521; Neil Tenant, Deflationism and the Gödel Phenomena, Mind 111, 551–582; and J. Ketland, Reply to Tennant, Mind 114, 75–88. The issue here concerned specifically the “thickness” of the notion of truth required for something like this argument to go through.
- 7.
The collection of sentences that Ag can prove to be true.
- 8.
Some Basic Theorems on the Foundations of Mathematics and Their Implications, in Gödel, Collected Works, III, Unpublished Essays and Lectures ed. by S. Feferman et al., Oxford University Press, 1986, p. 309.
- 9.
An idea subsequently vigorously defended by Hilary Putnam in, for example, Mathematical Truth, in Matheamtrics, Matter and Method, Philosophica Papers, v. 1 (1975).
- 10.
Charles Parsons has pointed out that it is not clear from Gödel’s writings in this period that talk about ‘perception’ of concepts is not to be taken metaphorically (see Platonism and Mathematical Intuition in Kurt Godel’s Thought, Bulletin of Symbolic Logic 1, 1 (1995)). In the 1960s, Gödel came to subscribe to Husserl’s phenomenological analysis of intentional consciousness, which makes available a more general characterization of the sort of epistemic access he was attempting to describe. It seems likely that Gödel would have accepted an analysis of ‘perception’ of concepts in terms of the noesis/noema correlation, on which concepts as ell as ordinary perceptual objects, are presented in a potentially infinite multiplicity of noetic acts through a corresponding multiplicity of noematic contents. For Gödel the (second-order) noema of the concept of set has an inexhaustibility analogous to that of the full noema of a concrete object. Gödel’s remarks suggest the possibility of novel applications of Husserl’s account of the intuition of concepts, and also some possible extensions of it. See Tieszen, After Gödel (Oxford University Press, 2013), for more on the Gödel–Husserl connection.
- 11.
What follows is an adaptation of self-referential phenomena in non-monotonic systems in my Self-Reference and Incompleteness in Non-Monotonic Structures, Journal of Philosophical Logic 23:4 (1994).
- 12.
For 1 see R. Jeroslow, Experimental Logics and Δ2 Theories, Journal of Philosophical Logic 4 (1975), 253–267;. For 2, 3 see McCarthy op. cit. A result related to 3 ois proved in P. Kugel, Induction Pure and Simple, Information and Control 35, 4 (1977) 276–336. The terminology of McCarthy (1994) and Jeroslow (1975) differs slightly from the above. In particular in McCarthy (1994) a frame is called a ‘non-monotonic structure’ and ‘indefeasible’ appears as ‘stable’. A sentence is said to be assertable in a non-monotonic structure if every state has an extension at which the sentence is stable. A non-monotonic structure is first-order closed if the collection of sentences assertable in it is first-order closed. In the present treatment, any sentence assertable in a non-monotonic structure is treated as potentially axiomatic.
- 13.
On both points, see Kripke, Wittgenstein on Rules and Private Language, Harvard University Press, 1982.
- 14.
I assume that the axioms of Q are evident if anything is. The consistency condition in conjunction with the provability of the instances of B ensures that any projection of the negation of such in instance is unstable. Karl may even from time to time counter-project (a). Consider a true properly Π2 statement of the form (a). Then the function f(n) = μkA(n,k) is not majorized by any primitive recursive function; else, the variable y in (a) can be primitive recursively bounded, so that (a) equivalent to a Π1 form. For a particular n, Karl may infer ¬∃yAn,y) inductively by reference to a long run of k’s such that ¬A(n,k) and thus temporarily project the negation of (a). But such a projection is never stable: eventually an integer k will be found such that the sentence A(n,k) is verified in Q. Again, if Karl’s inductive method is not hopelessly timid, Karl will frame the metainduction that such a projection is never stable and that the function f is in fact everywhere defined but not majorized by any p.r. function.
- 15.
The label ‘Maximize’, along with a number of important arguments for the position it represents, are due to Maddy, Naturalism in Mathematics, Oxford University Press, 1997.
- 16.
R.B. Jensen, Inner models and large cardinals, Bulletin of Symbolic Logic, 1 (1995): 393–407, p. 398.
- 17.
Jensen, op. cit. p. 400.
- 18.
Putnam, op. cit., E.M. Gold, Limiting Recursion, J. Symbolic Logic 30: 27–48 (1965), P. Kugel, Induction Pure and Simple, Information and Control 33: 276–336 (1977).
- 19.
See fact 3 of Sect. 5 and the references there.
- 20.
McCarthy (1994), Corollary 9.
- 21.
Penrose ref.
- 22.
Some basic theorems on the foundations of mathematics and their implications, op. cit.
- 23.
The argument from soundness, in its simplest form, is that “the axioms of T are true, the rules of inference preserve truth, so all the theorem of T are true. But ‘1 = 0’ is not true, so it is not a theorem of T. So T is consistent.”
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McCarthy, T. (2018). Normativity and Mechanism. In: Hellman, G., Cook, R. (eds) Hilary Putnam on Logic and Mathematics. Outstanding Contributions to Logic, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-96274-0_7
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