“NOMINALISM” AND “REALISM”

Nominalism is a large subject. In our book (Burgess and Rosen 1997) my colleague Gideon Rosen and I distinguished a negative or destructive side of nominalism, which tells us not to believe what mathematics appears to say, from a positive or reconstructive side, which aims to give us something else to believe instead. We noted that there were a few nominalists who contented themselves with the negative side, conceding that mathematics is useful, insisting that what it appears to say is not true, and letting it go at that, without attempting any reconstrual or reconstruction of mathematics. We expressed some surprise that there were not more such destructive nominalists, since as compared with reconstructive nominalism, destructive nominalism has what Russell in another context called “the advantages of theft over honest toil”; and if nothing else was clear from the work of Hartry Field, Charles Chihara, Geoffrey Hellman, and other reconstructive nominalists whose work we surveyed, it was clear that the amount of honest toil that would be required for a nominalistic reconstrual or reconstruction of mathematics would be quite considerable.

Today, a couple of years after publication, it is beginning to seem that the main achievement of our book will have been to provide a decent burial for the hard-working, laborious variety of nominalism.