Alan Turing pranks Wittgenstein

In The Inevitable Gift Shop (which I've blogged about before), the author Will Eaves recalls a Cambridge encounter in 1936 between Alan Turing and Ludwig Wittgenstein, a conversation about error and common sense in which the former appears to wrong foot the latter by offering a decidability paradox: 'One can never know that one has not made a mistake'.

Eaves glosses that thus:

If this is true, then one cannot in general know if one has made a mistake or not, in which case the sentence may be false; but if it is false, then one can know in general if one has made a mistake or not, and the sentence becomes decidable, which means it could be true.

So if the statement is correct then it is false, and if the statement is incorrect then it is demonstrably true. You may, as I did, need a moment to mull this over . . .

What are we to make of the once-commonplace author disclaimer (in works of non-fiction), following the usual acknowledgements to colleagues and supporters and family and publisher and proofreaders, that "any remaining mistakes are of the author's making"?

How can the author, how can any author, be so sure? If there are errors which he knows to be his own, why not correct them before publication? And if he doesn't correct them, and therefore hasn't noticed them, how does he know for certain they are his?  (I seem to remember the great Brian O' Nolan, aka Flann O'Brien, made an observation along these lines many years ago, perhaps in his celebrated Irish Times column Cruiskeen Lawn).

But back to the Turing paradox. My mind is easily made giddy by even the mildest of paradoxes (if there is a taxonomy of paradox ranging from the 'so-what?' to the 'well-I'll be damned'). Young children are literalists and entirely immune to paradox. I recall showing my son (aged five) that image of a duck/rabbit used by Wittgenstein in Philosophical Investigations and asking hime: "Is it a duck or a rabbit?" He replied without a moment's hesitation: "Yes. It's a duck or a rabbit".

A drabbit. Or a ruck.