A footnote to my post, Logic disappearing over the horizon. I’ve just been reading Stephen Simpson’s “Unprovable Theorems and Fast-Growing Functions” (an introductory piece in the 1987 AMS Contemporary Mathematics Logic and Combinatorics volume that contains some important papers on provably computable functions — it is a pity that Simpson’s very helpful and accessible survey isn’t more readily available, e.g. on his website). I was struck by this remark:
Like most good research in mathematical logic, the results which I am going to discuss had their origin in philosophical problems concerning the foundations of mathematics.
And that’s right: the most interesting work in mathematical logic is bound up with problems and projects of a more philosophical kind concerning the foundations of mathematics. All the more worrying, then, the seeming trend I was remarking on for logic courses to be less and less available even to graduate philosophy students. If the wonderfully fruitful long dialogue since Frege between philosophers and mathematicians (or often, between the philosophical and mathematical sides of the same individual) is to continue, then some philosophers at any rate do need to be logically well-educated!