The philosophy of mathematics is philosophy in a pure state, stripped of all worldly appendages; austere; philosophy without the sugar-coating of a pretence to Relevance to Life; but also heady, perhaps best taken in short sips. Much of the present paper was written for a conference entitled "'Philosophy of Mathematics Today'", for which I was invited by Matthias Schirn to reflect on two papers that I published decades ago2 and to comment on how the issues discussed in them had fared in the intervening years.
Rereading them at this distance, I was surprised to see how much they were the product of a particular time and soon found myself plunged into the philosophical atmosphere from which they had sprung. The papers in question have chequered histories. 'What Numbers Could Not Be' (henceforth 'WNCNB') was written at the invitation of Max Black for possible inclusion in his volume Philosophers in America ( 1965). The book-jacket describes the essays selected for the volume as 'representing the best of the lively and penetrating work of the most talented younger American philosophers', an ambition that the wily Max had managed to convey in his letter of invitation, making it very hard to refuse to submit an entry. But, a representative of 'the best of the lively and penetrating work of the most talented younger American philosophers' turned out to be what WNCNB could not be--it didn't make the cut (happily for it, I think, since a number of excellent papers were long entombed there, including Charles Parsons' 'Frege's Theory of Number').
"'Mathematical Truth'" (henceforth MT) was written in December-January of 1967-8, sent to a couple of trusted friends, and put away in a drawer. Although persuaded of the thrust of the paper, I was very unhappy with its detail. Despite that, having nothing else in my drawer, it was what I read when invited to give talks. It was a lead balloon, meeting with extraordinary and uniform silence on the part of every audience. Through those trusted (but not trustworthy) friends, it also received a wide underground circulation in manuscript form, which is how it remained until I was invited to make it the topic of a symposium of the American Philosophical Association in December 1973, roughly six years after writing it. Despite____________________