Thursday, June 07, 2007

Church’s Thesis 10: Precision and pretension

So, we're halfway through my blogview of Church’s Thesis After 70 Years edited by Adam Olszewski, Jan Wolenski and Robert Janusz: I've commented on eleven papers, and there are eleven to go. I'm afraid that so far I've not been wildly enthusiastic: the average level of the papers is not great. But there are cheering names yet to come: Odifreddi, Shapiro, Sieg. So I live in hope ...

But the next paper -- Charles McCarty's 'Thesis and Variation' -- doesn't exactly raise my spirits either. The first couple of sections are pretentiously written, ill-focused, remarks about 'physical machines' and 'logical machines' (alluding to Wittgenstein). The remainder of the paper is unclear, badly expounded, stuff about modal formulations of CT (in the same ball park, then, as Horsten). Surely, in this of all areas of philosophy, we can and should demand direct straight talking and absolute transparency: and I've not the patience to wade through authors who can't be bothered to make themselves totally clear.

At least the next piece, five brisk sides by Elliott Mendelson, is clear. He returns to the topic of his well known 1990 paper, and explains again one of his key points:

I do not believe that the distinction between “precise” and “imprecise” serves to distinguish “partial recursive function” from “effectively computable function”.

To be sure, we offer more articulated definitions of the first notion: but, Mendelson insists, we only understand them insofar as we have an intuitive understanding of the notions that occur in the definition. Definitions give out at some point where we are (for the purposes at hand) content to rest: and in the end, that holds as much for “partial recursive function” as for “effectively computable function”

Mendelson's point then is that the possibility of establishing the 'hard' direction of CT can't be blocked just by saying that the idea of a partial recursive function is precise, the idea of an effectively computable function is isn't, so that there is some sort of categorial mismatch. (Actually, though I take Mendelson's point, I'd want stress a somewhat different angle on it. For CT is a doctrine about the co-extensiveness of two concepts. And there is nothing to stop one concept having the same extension as another, even if the first is in some good sense relatively 'imprecise' and the second is 'precise' -- any more than there is anything to stop an 'imprecise' designator like “those guys over there” in the circumstances picking out exactly the same as “Kurt, Stephen, and Alonzo”.)

As to the question whether the hard direction can actually be proved, Mendelson picks out Robert Black's “Proving Church’s Thesis”, Philosophia Mathematica 2000, as the best recent discussion. I warmly agree, and I take up Robert's story in the last chapter of my book.

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