Tuesday, June 12, 2007

Church’s Thesis 12: Kreisel, Church

I’m going to pass over the next three papers in the Olszewski collection pretty quickly, for various reasons.

First, there’s a piece on “Analog computation and Church’s Thesis” by Jerzy Mycka: I just don’t know anything about this stuff, and am certainly not in a position to comment on this piece, though it looks decent enough.

Then there is a paper by Piergiorgio Odifreddi on “Kreisel’s Church”. Now, I find Kreisel’s writings intriguing and irritating in roughly equal measure (all sorts of interesting things seem to be going on, but he just can’t or won’t write flat-footed technical exposition and direct, transparent, plain-spoken, philosophical commentary). So I’d hoped that Odifreddi might take up one or two of Kreisel’s themes from his discussions of CT and give them a careful development and exploration -- for Odifreddi's wonderful book on Recursion Theory shows how clear and judicious he can be. But regrettably, he does something quite different: he takes us on a lightening survey of Kreisel’s many relevant articles, with a lot of quotations and some references to related later work. But if you were puzzled by Kreisel before, you’ll stay pretty puzzled -- though you’ll have a longer bibliography!

Next, Adam Olszewski contributes a short piece on “Church’s Thesis as interpreted by Church”. Here Olszewski does at least pick up on the point that I noted that Murawski and Wolenski slurred over without comment. CT is nowadays usually taken to be a claim about the co-extensiveness of the two notions of effective computability and recursivess; but the Founding Fathers were wont to talk of the notions being identical, or of one notion being a definition of the other. Church in 1936 himself uses both identity talk and definition talk. But actually, it isn’t too likely that -- at that early date -- Church had a clearly worked out position in mind on the status of the correlation he was proposing between computability and recursivess, and I don’t think we can read too much into his choice of words here. The headline is that Church, at least originally, seemed to take the modest view of the correlation as having something like the status of a Carnapian explication, while Turing thought, more boldly, that it could be forced on us by an analysis of the very notion of a computation. This is familiar stuff, however, and Olszewski’s brief remarks don’t in any way change the familiar picture.

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