Greg Restall has put a very nice paper online, called ‘Anti-realist classical logic and realist mathematics’. I’ve always been tempted by logicism in the very broadest sense: but Greg’s critics, of course, will say that he’s just smuggled the rabbit into the hat before pulling it out again. Still, his piece is nicely thought-provoking.
Oddly, given his multiple-conclusion logical framework, Greg doesn’t mention Shoesmith and Smiley’s great book in his biblio. (Very regrettably, it had the bad luck to be published in 1978, around when a number of publishers used early computers to print books with what looked like typewritten pages. The first edition of Fogelin’s fine book on Wittgenstein suffered the same fate. And in both cases, I think the repellent and amateurish look of the results was enough to put readers off and stop the books making the impact they deserved at the time. At least Fogelin got a properly printed second edition. But Shoesmith and Smiley has gone out of print, and seems widely forgotten.)
Thanks for the nice words. They mean a lot, coming from you.
The paper is a rough draft and it needs more work before it’s anywhere near complete. I should mention S&S — which has been influential in my thinking. My only excuse is that it is in the transitive closure of the bibliography. It’s cited and discussed in my “Multiple Conclusions” which I cite in that paper.
I must also agree with my critics here: Of course there’s a rabbit in the hat before it’s pulled out. (The mathematics doesn’t come from nowhere.) It’s at most a semi-logicism, designed to bring to us classical-logic-loving folks the Nice Features of second order arithmetic defined proof-theoretically. Or so I hope… I’ve got to get back to the paper, but I’m brushing up what I think about the proof theories of propositional and predicate logics before I get to interesting theories like arithmetic.