Abstract
In this chapter we refute one of the core parts of computationalism, as this doctrine is defined in Chapter 1: namely, Church’s Thesis. More precisely, we refute the Church- Turing Thesis, the view, put roughly for now, that that which is algorithmic in the intuitive sense can be identified with what a (standard) Turing machine can accomplish. As you will see, our refutation makes crucial use of the formal terrain above the Turing Limit (recall, yet again, Figure 1.1). The refutation owes a great debt to Elliot Mendelson, who, in a widely affirmed paper in the Journal of Philosophy (1986), challenges what he rightly calls the “standard conception” (Mendelson 1986, p. 230) of Church’s Thesis (CT) — the conception being that CT is unprovable. Mendelson got Bringsjord thinking about CT in earnest; he provided the stimulus that eventuated in this chapter. However, our core argument owes much to many other thinkers who have offered critiques, and our case has in the process — by our lights, anyway — grown progressively stronger.
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Footnotes
Recall our remarks, in the Preface, about Turing (1936)’s “computists” and Post (1944)’s “workers,” humans whose sole job was to slavishly follow explicit, excruciatingly simple instructions.
WT identifies the intuitive notion of limit with the standard e-δ definition. Mendelson thinks that these four theses are just some among many such “theses.’’ He mentions “the notion of measure as an explication of area and volume, the definition of dimension in topology, the definition of velocity as a derivative, the definition of logical implication and logical equivalence in first-order logic, and the definitions of circle, triangle, interior of an angle, and many other geometric concepts” (Mendelson 1986, p. 232).
Personal communication, December 14, 1993.
Incidentally, the reconstruction probably has fatal problems, as Folina (1993) points out. After all, is what Mendelson calls a proof here a proof? One common conception — indeed, probably the dominant conception — of a proof is of a transformation in some formal system. Yet Mendelson says about his proof: “The fact that it is not a proof in ZF or some other axiomatic system is no drawback; it just shows that there is more to mathematics than appears in ZF” (Mendelson 1986, p. 233). (Remember this quote later when we consider Mendelson’s rejection of Kalmár’s argument against CT because in part it falls outside any standard formal system.) A lot of thinkers will balk at this. As Folina (1993) notes, many will diagnose the situation by saying that what Mendelson has shown is that there is more to mathematics than proofs. (This is something we’ve known all along, of course.) Moreover, if Mendelson’s reasoning isn’t a proof, then what is it? If it’s merely a precise, compelling argument connecting an intuitive notion with a formal one, then it shows something we knew to be true all along.
Relevant here is Hofstadter’s LETTER SPIRIT program, which generates fonts from the first few letters in the font in question. For an argument that this program, and others, aren’t really creative, see (Bringsjord, Ferrucci & Bello 2001).
A search for coverage of this concept in standard texts about cognition — e.g., (Ashcraft 1994) and (Stillings et al. 1995) — turns up nothing whatever.
What argument could be mustered for ignoring beauty in the context of attempts to reduce cognition to computation, or to build an artificial agent capable of behaviors analogous to human ones typically taken to involve beauty? We envisage an argument running parallel to the one John Pollock (1995) gives for ignoring human emotions in his attempt to build an artificial person. Pollock’s view, in a nutshell, is that human emotions are in the end just “time savers;” with fast enough hardware, and clever enough algorithms, artificial persons could compute the need to quickly flee (say) a lion, whereas we take one look and immediately feel a surge of fear that serves to spark our rapid departure.
An insightful review of this book has been written by Tom Trabasso (1996).
From page 592 of (Charniak & McDermott 1985). The story is studied in the context of attempts to resolve pronouns: How do we know who the first occurrence of ‘He’ refers to in this story? And how do render the process of resolving the pronoun to Jack as a computational one?
This intelligent objection is originally due to Michael McMenamin (1992), though a number of thinkers have conveyed its gist to us.
Those unfamiliar with Eco’s non-fiction work, might start with his surprising reasons for finding Ian Fleming’s 007 (James Bond) series to be very interesting; see “Chapter Six: Narrative Structures in Fleming,” in (Eco 1979).
Here is one example from (Pollock 1995): Pollock’s OSCAR system is designed so as to constantly update that which it believes in response to the rise and fall of arguments given in support of candidate beliefs. What constitutes correct reasoning in such a scheme? Pollock notes that because a TM with an ordinary program can’t decide theorems in first-order logic (the set of such theorems isn’t Turing-decidable), answering this question is quite tricky. He ingeniously turns to super-computation for help: the basic idea is that OSCAR’S reasoning is correct when it generates successive sets of beliefs that approach the ideal epistemic situation in the limit. This idea involves AH, as Pollock explains.
Boolos and Jeffrey, for example, in their classic textbook Computability and Logic (1989), provide a sustained discussion of CT — and take pains to leave the reader with the impression that CT can be overthrown.
Perhaps we should mention here something that students of CT and its history will be familiar with, viz., given an intuitionistic interpretation of ‘effectively computable function,’ CT can be disproved. The basic idea is to capitalize on the fact that any subset of N is intuitionistically enumerable, while many such sets aren’t effectively enumerable. [A succinct presentation of the disproof can be found on page 592 of Nelson (1987).] The main problem with such attacks on Church’s Thesis, of course, is that they presuppose (certain axioms of — see e.g., Kreisel 1965, 1968) intuitionistic logic, which most reject.
The original proof can be found on page 741 of (Kleene 1983).
It will not be necessary to present here the formal extension of computability with number-theoretic functions to computability with functions over the reals. For the formal work, see, e.g., (Grzegorczyk 1955, 1957).
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Bringsjord, S., Zenzen, M. (2003). Supermentalism and the Fall of Church’s Thesis. In: Superminds. Studies in Cognitive Systems, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0283-7_4
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