I mentioned a few posts ago the collection Church’s Thesis After 70 Years edited by Adam Olszewski et al. Since the editors helpfully refrain from suggesting a sensible reading order, I’m just going to dive in and read the contributed papers in the order they are printed (they are arranged alphabetically by the authors’ names). And to keep my nose to the grindstone, I’ve promised myself to post comments and thoughts here — so it will be embarrassing to stop doing my homework! Here goes, then, starting with Darren Abramson, “Church’s Thesis and Philosophy of Mind”.

Abramson identifies “the Church-Turing Thesis” with the claim “[…] no human computer, or machine that mimics a human computer, can out-compute a universal Turing machine”. That doesn’t strike me as a terribly helpful move, for it runs together two claims, namely (i) no human computer can (in a reasonable sense) compute something that is not effectively computable (by a finite, step-by-step, algorithmic process), and (ii) whatever is effectively computable is Turing-computable/recursive. In fact, in my Gödel book, I call just (ii) “the Church-Turing Thesis”. But irrespective of the historical justification for using the label my way (as many do), this thesis (ii) is surely to be sharply separated from (i). For a start, the two claims have quite different sorts of grounds. For example, (i) depends on the impossibility of the human performance of certain kinds of supertask. And the question whether supertasks are possible is quite independent of the considerations that are relevant to (ii).

I didn’t know, till Abramson told me, that Bringsjord and Arkoudas have argued for (i) by purporting to describe cases where people do in fact hypercompute. Apparently, according to them, in coming to understand the familiar pictorial argument for the claim that lim n → ∞ of 1/2^n is 0 we complete an infinite number of steps in a finite amount of time. Gosh. Really?

Abramson makes short shrift of the arguments from Bringsjord and Arkoudas that he reports. Though I’m not minded to check now whether Abramson has dealt fairly with them. Nor indeed am I minded to bother to think through his discussion of Copeland on Searle’s Chinese Room Argument: frankly, I’ve never felt that that sort of stuff has ever illuminated serious issues in the philosophy of mind that I might care about. So I pass on to the next paper ….

Aatu KoskensiltaI agree that “[…] no human computer, or machine that mimics a human computer, can out-compute a universal Turing machine” is a rather unfortunate phrasing for the Church-Turing thesis. It is either an empirical claim about the capabilities of actual humans — which is not something I find very relevant in context of the thesis — or a confused paraphrase of the purely conceptual thesis that mechanical computability is (extensionally) equivalent to Turing-computability.

Concerning the latter alternative it might be good idea to consider other similar rephrasings of mathematical and conceptual claims. For example, it is often said that if Goldbach’s conjecture is false we could in principle find this out. In reality there is obviously no guarantee of anything like that unless we interpret “we could in principle find this out” as a somewhat misleading expression of the purely mathematical fact that deciding of any particular natural whether it’s a counterexample to Goldbach’s conjecture is a recursive task. It seems to me that in context of fundamental conceptual questions and investigations such rephrases are liable to cause only confusion and should be avoided, even if in many other contexts they’re totally innocent.

How’s that for a collection of platitudes.