Abstract
We ended the last chapter by noting that though Roger Penrose’s case against computationalism fails, his intuitions about the hypercomputational nature of mathematical reasoning may nonetheless be correct. Specifically, as we show in the present chapter, the infinitary nature of some of this reasoning may be part of what makes us superminds. As such, this chapter provides a much more direct route than the Gödelian one Penrose has long been fighting to successfully take.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Footnotes
A nice paper on this issue is (Hoffman 1993), wherein (put roughly) the authors argue for the view that experts simply “see” things novices don’t. In addition, there seems to be an emerging consensus in education that there is a genuine distinction to be made between “shallow” versus “deep” learning; see (Graesser, VanLehn, Rose, Jordan & Harter 2001, Aleven, Koedinger & Cross 1999).
We should mention that the example given at end of the preceding chapter as a possible example of infinitary/hypercomputational mathematical reasoning is but one out of many that could be given. For example, it strikes us as entirely possible that the development of infinitesmals happened on the strength of such reasoning. For coverage of this history, see (McLaughlin 1994, Davis & Hersh 1972, Nelson 1977). Leibniz seems to us to have engaged in infinitary reasoning (at the intuitive level; recall the distinction between intuition and ingenuity made in the previous chapter) when discussing infinitesmals.
See (Bringsjord 1991) for a discussion of the consequences of this fact for AI.
The capabilities added must, however, be expressible in the language of set theory. If one considers a physical Turing machine, then there are perhaps ways of “souping up” such machines so that they can process uncomputable functions. See Chapter 4 in the present volume, and (Bringsjord 2001b).
The interested reader can consult an octet of books we find useful: For broad coverage of the basic material, see (Lewis & Papadimitriou 1981, Ebbinghaus et al. 1984, Boolos & Jeffrey 1989, Hopcroft & Ullman 1979). For a nice comprehensive discussion of computability theory that includes succinct coverage of uncomputability, including the Arithmetic Hierarchy, see the book from which we drew heavily in Chapter 1: (Davis et al. 1994); see also the difficult but rewarding (Soare 1980). (Partee et al. 1990), as mentioned in Chapter 1 (when we discussed the Chomsky Hierarchy), contains a very nice discussion of the Chomsky Hierarchy. And, of course, there’s always the classic (Rogers 1967).
This chapter is aimed at a multidisciplinary audience assumed to have familiarity with but the rudiments of logic and AI. So this isn’t the place to baptize readers into the world of cardinal numbers. Hence we leave the size implications of the subscripts in Lω1ω, and other related niceties, such as the precise meaning of ω, to the side. For a comprehensive array of the possibilities arising from varying the subscripts, see (Dickmann 1975).
A recent treatment of the issues here can be found in Smullyan’s (1992) recent book on Gödel’s incompleteness results. Many logicians have a general notion of Gödel’s first incompleteness theorem, but few know that Gödel showed that there is a formula φ(y), with one free variable y, such that φ(1). φ(2), φ(3),…, φ(n),…are all provable in Peano Arithmetic (PA), while the sentence ∀yφ(y) isn’t. This phenomenon — called ω-incompleteness by Tarski — can be remedied by invoking the system PA+, which contains the —-rule (sometimes also called Tarski’s rule or Carnap’s rule) allowing one to infer ∀yφ(y) from the infinitely many premises φ(1),φ(2), φ(3),…, φ(n),…. This rule of inference, which we discussed earlier, again, is at the heart of the present chapter.
The introduction in (Moore 1990) is one of the best synoptic accounts of these and other paradoxes.
Please note that our rebuttal doesn’t in the least conflate object theory with metatheory. We are in fact invoking this very distinction — but we’re pointing out that the metatheory in question (unsurprisingly) deploys some of the very same infinitary constructions as seen in Lω1ω. This seems utterly undeniable — and so much the worse for those (e.g., Ebbinghaus et al. 1984) who believe that “background” logic/mathematics is fundamentally first-order.
A fully formal proof in F (the system of natural deduction introduced by Barwise & Etchemendy 1999) is shown in Figure 3.9. The predicate letter H is of course used here for ‘helped.’
F is the natural deduction system presented in in Barwise and Etchemendy’s Language, Proof, and Logic (1999). In such a system, there is always a rule for introducing a logical connective, and a rule for eliminating such a connective.
The situation is different than chess — radically so. In chess, as explained in (Bringsjord 1998a), we knew that brute force could eventually beat humans. In reasoning, brute force shows no signs of exceeding human reasoning. Therefore, unlike the case of chess, in reasoning we are going to have to stay with the attempt to understand and replicate in machine terms what the best human reasoners do. See prescription P4 in the final chapter.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Bringsjord, S., Zenzen, M. (2003). The Argument from Infinitary Reasoning. In: Superminds. Studies in Cognitive Systems, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0283-7_3
Download citation
DOI: https://doi.org/10.1007/978-94-010-0283-7_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-1095-8
Online ISBN: 978-94-010-0283-7
eBook Packages: Springer Book Archive