Monday, August 21, 2006

Joined up thinking and noisy cafés

Jacob Plotkin has e-mailed to point out a stupid mistake just five pages into the previous version(s) of the Gödel book. I'd given a really, really bad reason for saying (what is true) that the incompleteness of arithmetic entails the incompleteness of the theory of rational fields. Of course, it's not enough for Gödelian incompleteness to carry over to a theory T that T can define 0, 1, 2 and so on (and knows about adding and multiplying these). T has got to be able to define a predicate Nat applying to the numbers so that T can replicate numerical quantifications. Only that way can T count as e.g. embracing Robinson arithmetic, and hence be incomplete-if-consistent for Gödelian reasons. Well, Q, the standard theory of rational fields, can define a suitable predicate Nat, so incompleteness does apply. But that is not at all obvious. In fact, defining Nat in Q was part of the work that Julia Robinson got her PhD for (see her JSL 1949 paper).

Now I kind of knew all that, but it didn't stop me writing something that completely ignored it. And it stayed in a number of successive drafts. These kinds of cognitive glitches -- these failures in joined up thinking -- are very odd, and maddeningly annoying when you succumb to them. (I once saw my wife, who'd spent some of the afternoon making a large pot of chicken stock, very attentively pouring the stock away down the sink, carefully saving the boiled bones and vegetables ...)

Another psychological oddity: why do I often find it easier to read in my favourite noisy Italian café (Savino's, since you ask) with Italian radio blaring and a lot of comings and goings, than in a quiet library which can be all too conducive to sleepiness. I don't think it is just the supply of espresso. Presumably it is something to do with the noise and bustle keeping part your brain alert for signs of danger and threat in the background, and so stopping processing from shutting down ...

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