FOM: Putnam on mathematics and possibility

Charles Parsons parsons2 at
Tue Jan 20 08:40:53 EST 1998

In his posting of Monday, January 19, 12:11, Reuben Hersh seems to
attribute to Hilary Putnam a reduction of mathematics to "physical
possibility." He is probably thinking of Putnam's 1967 article "Mathematics
without Foundations," reprinted in Putnam's collection _Mathematics,
Matter, and Method_ (Cambridge 1975). There Putnam sketched a translation
of the language of set theory into a modal language, where the idea was to
replace the existence of mathematical objects with, roughly, the
possibility of structures. But Putnam made clear that the possibility
involved was not physical but distinctively mathematical.

I have never seen how to work out Putnam's sketch, but the idea has been
pursued by others, most fully by Geoffrey Hellman in _Mathematics without
Numbers_ (Oxford 1989). How much of a reduction one gets is something
philosophers argue about. (My own views on the matter are in "The
structuralist view of mathematical objects" (Synthese 84, 1990), reprinted
in W. D. Hart's collection _The Philosophy of Mathematics_ (Oxford 1996).)
I think there would be general agreement that there could be no reduction
of set theory to physical concepts, even if physical modalities are

Charles Parsons
Edgar Pierce Professor of Philosophy, Harvard University
Research interests: philosophy, especially of logic and mathematics,
historical figures in the subject from Kant to Goedel, in earlier years
proof theory

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