A day that didn’t quite go to plan in a couple of ways, but still, three good logical things happened.
First, I’d been given the task of updating and expanding the logicky part of the entrance test done by undergraduates applying to read philosophy at Cambridge. It would be giving the game away to say very much about this. But I did have quite a bit more fun than I was expecting trawling around to see how others — like USA grad schools — manage these things, and putting together some suitably testing questions. (I’m not sure I’d have ever got in to Cambridge with all these new fangled tests, and interviews to find out if you are a well-rounded human being: in my day I just did a lot of nasty problems in projective geometry very fast, and bingo …)
Second, it seems that — thanks to Thomas Forster — my plan to learn some more model theory by running a reading group with some hard core mathmos joining in seems as though it will come off next term: that should be terrific. The default suggestion is that we do the shorter Hodges (though the Marker book looks a possibility too).
And third, not least, it was Logic Seminar day — the highpoint of the academic week really. At the moment, at Michael Potter’s instigation, we are thinking about later Dummettian arguments against realism in mathematics. Or rather Michael and some of our grads are thinking, and I’m trying to keep up. Today we were looking at Peter Sullivan’s contribution to the new “Schillp” volume on Dummett, where he seeks to locate in Frege: Philosophy of Mathematics a really rather simple argument for anti-realism. Applied to arithmetic, we have Premise One: The logicist claim that arithmetic is broadly analytic, its true claims being made true by the character of our concepts. Premise Two: what is given in our concepts does not suffice to fix the truth-value of every claim in the language arithmetic . So there is nothing to fix it that every such claim is determinately true or false.
This is an argument against realism in mathematics which (a) doesn’t threaten to sprawl in ways that are difficult to handle to become a general anti-realism, and (b) doesn’t depend on problematic claims about indefinite extensibility. Peter Sullivan certainly seems to show that the argument is indeed a thread running through the later Dummett, and his exploration is very illuminating (even if I think I lost the plot a bit towards the end of this long paper). I learnt a lot from the discussion. Great stuff.