Church’s Thesis 3: Constructivism, informal proofs

The third, short, paper in the Olszewski collection is by Douglas S. Bridges — the author, with Fred Richman, of the terrific short book Varieties of Constructive Analysis. The book tells us a bit about what happens if you in effect add an axiom motivated by Church’s Thesis to Bishop-style constructive analysis. This little paper says more on the same lines, but I don’t know enough about this stuff to know how novel/interesting this is.

The fourth paper is “On the Provability, Veracity, and AI-Relevance of the Church-Turing Thesis” by Selmer Bringsjord and Konstantine Arkoudas. At least these authors get it right about what the core Thesis is: a numerical function is effectively computable if and only if it is Turing-computable. But that’s about as good as it gets. The first main section is a bash against Mendelson’s argument against “the standard conception of the thesis as mathematically unprovable”. Now, although I am sympathetic to Mendelson’s conclusion, I’d want to argue for it in a rather different way (and do so in the Gödel book). But Bringsjord and Arkoudas’s objections just seem badly point-missing about the possibility of a Mendelsonian line. Their argument (p. 69) depends on a bald disjunction between proofs in formal systems and what they call “empirical evidence” for CTT. But of course, tertium datur. Take, for example, the familiar theorem that there are effectively computable functions which aren’t primitive recursive. I’m not being tendentious in calling that a theorem — that’s how the textbooks label the result. And the textbooks, of course, give a proof using a diagonalization argument. And it is a perfectly good proof even though it involves the informal notion of an effectively computable function (the argument isn’t a fully formalizable proof, in the sense that we can’t get rid of the informal notions it deploys, but it doesn’t just give “empirical” support either). Now, the Mendelsonian line is — I take it — precisely to resist that dichotomy formal/formalizable proof vs mere quasi-empirical support and to remind us that there are perfectly good informal proofs involving informal concepts (and Mendelson invites us to be sceptical about whether the informal/formal division is in fact as sharp as we sometimes pretend). And the key question is whether it is possible to give an informal mathematical proof of CTT as compelling as the proof that not all effectively computable functions are primitive recursive. Reiterating the rejected dichotomy is of course no argument against that possibility.

That’s not a great start to the paper (though the mistake is a familiar one). But things then go further downhill. For much of the rest of the paper is devoted to a discussion of Bringsjord’s claim that membership of “the set of all interesting stories” (!!) is effectively decidable but not recursively decidable (Gödel number the stories and we’d have a counterexample to CTT). And what, according to Bringsjord, is the rationale behind the claim that members of that set is effectively decidable? “The rationale is simply the brute fact that a normal, well-adjusted human computist can effectively decide [membership]. Try it yourself!” Well, you might be able to decide in many cases (always? just how determinate is the idea of an interesting story?): but who says that it is by means of implementing a step-by-step algorithm? Effective decidability is a term of art! It doesn’t just mean there is some method or other for deciding (as in: ask Wikipedia or use a cleverly tuned neural net). It means that there is an algorithmic procedure of a certain sort; and in trying to judge whether a story is interesting it most certainly isn’t available to inspection whether I’m implementing a suitable algorithm. This whole discussion just seems badly misguided.